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Study of gaseous detectors for high energy physics
experiments
A thesis Submittedin Partial Fulfilment of the Requirements
for the Degree of
MASTER OF SCIENCE
by
Sumanya Sekhar Sahoo
School of Physical Sciences
National Institute of Science Education and Research
Bhubaneswar
Abstract
Gas filled detectors are one of the oldest and most widely used radiation detectors
and based on the effects produced when a charged particle passes through a gas.
Many ionization mechanism arise in gases and over the years these have been studied
and exploited in the detector. In this particular work we have studied a few aspects
of these different phenomena. We have described here the principles of operation of
the gas ionization chamber focussing on the most relevant mechanisms involved in the
detection process. This knowledge is necessary to understand, simulate and optimize
the performance of the proposed or any existing detection system. The simulation
procedure is now-a-days widely utilize in designing such detectors. Starting with the
basic principle of operation of gas detectors, we first discuss the different regions of
operation of gas detectors. Following which, the production of primary and secondary
electrons, transport of charged particle in gases, the electron multiplication process,
the required characteristics of the gas mixture have been described. Then we illustrate
the basic features of some gas detectors. We also give a brief idea about the simulation
and present some calculation utilizing various computer programmes.
ACKNOWLEDGEMENTS
I thank to Dr. Bedangadas Mohanty for his constant support, Dr. Purba Bhat-
tacharya for her detailed guidance and Dr. Saikat Biswas for the clarification of my
doubts when ever needed.
i
Contents1 Introduction 1
2 Basic Principles of interaction of radiation with matter 62.1 Interaction of radiation and matter . . . . . . . . . . . . . . . . . . . 6
2.1.1 Interaction of charged particles in a gas . . . . . . . . . . . . . 72.1.2 Interaction of neutral particles in a gas . . . . . . . . . . . . . 9
2.2 Ionisation Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Gaseous ionisation Detectors . . . . . . . . . . . . . . . . . . . 112.2.2 Choice of Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Different types of ionisation mechanism . . . . . . . . . . . . . . . . . 18
3 Energy loss of Charged Particles 203.1 Bethe-Bloch formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Photoabsorption ionization model . . . . . . . . . . . . . . . . . . . . 25
3.2.1 Cross Section for Ionizing Collisions . . . . . . . . . . . . . . . 253.2.2 Energy Loss Simulation . . . . . . . . . . . . . . . . . . . . . 28
4 Transport of charged particle in Gases 294.1 Drift of Charged Particle . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Microscopic Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.1 Drift of Electrons . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Drift of Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.4 Inclusion of Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . 364.5 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.6 Electric Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.7 Magnetic Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.8 Amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.8.1 Electron multiplication of the gas . . . . . . . . . . . . . . . . 444.8.2 The first Townsend coefficient . . . . . . . . . . . . . . . . . . 474.8.3 The Second Townsend Coefficient . . . . . . . . . . . . . . . . 48
4.9 Electron loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.9.1 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . 494.9.2 Electron Attachment . . . . . . . . . . . . . . . . . . . . . . . 50
5 Measurement of Ionization 525.1 Signals Induced on Grounded Electrodes, Ramos Theorem . . . . . . 525.2 Induced Signals in a Drift Tube . . . . . . . . . . . . . . . . . . . . . 54
6 Simulation of Gaseous detectors 586.1 Simulation tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.1.1 Garfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.1.2 MAGBOLTZ . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.1.3 Calculation of the electric field . . . . . . . . . . . . . . . . . . 676.1.4 Accuracy check of neBEM . . . . . . . . . . . . . . . . . . . . 71
ii
CONTENTS
7 Results and Discussion 747.1 Multi Wire Proportional Chamber . . . . . . . . . . . . . . . . . . . . 747.2 Gas Electron Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.2.1 Principle of operation . . . . . . . . . . . . . . . . . . . . . . . 777.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8 Summary and Conclusions 90
References 91
iii
List of Figures1.1 Various types of detection techniques . . . . . . . . . . . . . . . . . . 2
2.1 Schematic setup for a gaseous ionisation detector. . . . . . . . . . . . 122.2 I-V characteristics of Gas detectors. . . . . . . . . . . . . . . . . . . 132.3 Ionisation mechanisms considering Bohr’s model of an atom. . . . . . 19
3.1 Incoming fast particle of charge ze scattered by an atomic electronalmost at rest: for small energy transfer, the particle trajectory is notdeflected . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 An incoming fast particle of charge ze interacts with electrons at im-pact parameter between b and b+ db . . . . . . . . . . . . . . . . . . 22
4.1 Mobility variation inside an electron cloud travelling in the z direction. 434.2 Process of recombination and Ionisation. . . . . . . . . . . . . . . . . 50
5.1 Cylindrical proportional tube with outer radius b at voltage V0 andinner (wire) of radius a at voltage zero. . . . . . . . . . . . . . . . . . 55
5.2 Current signal according to Eq-5.19 (full line, left-hand scale) for t0 =1.25ns, b/a = 500, and q = 106∗e. Time integral of this pulse (Inducedcharge) as a percentage of the total (broken line, right-hand scale) . . 57
6.1 An overview of the different methods on quest of truth is explainedthrough a flow chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Variation of drift velocity of electron in different gas mixtures. . . . . 626.3 Variation of Townsend coefficient of electron in different gas mixtures. 636.4 Variation of attachment coefficient of electron in different gas mixtures. 636.5 Variation of longitudinal diffusion coefficient of electron in different gas
mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.6 Variation of Transverse diffusion coefficient of electron in different gas
mixtures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.7 Variation of longitudinal diffusion coefficient of electron inNe–CO2–N2
(90-10-5) with magnetic field. . . . . . . . . . . . . . . . . . . . . . . 656.9 Variation of transverse diffusion coefficient of electron in Ne–CO2–N2
(90-10-5) with magnetic field (Perpendicular to electric field). . . . . 666.8 Variation of transverse diffusion coefficient of electron in Ne–CO2–N2
(90-10-5) with magnetic field (parallel to electric field). . . . . . . . . 66
7.1 Schematic diagram of a MWPC read out and use of gating grid. . . . 757.2 Volatage and Electric field configuration of MWPC. . . . . . . . . . . 757.3 Simulating an Avalanche in MWPC. . . . . . . . . . . . . . . . . . . 767.4 Image of the GEM foil taken with an electron microscope. . . . . . . 787.5 Schematic diagram of a GEM detector’s readout. . . . . . . . . . . . 797.6 Basic GEM cell built using Garfield. . . . . . . . . . . . . . . . . . . 817.7 Voltage and Electric field along the z-axis passing through hole of a
GEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.8 Electron drift lines in GEM. . . . . . . . . . . . . . . . . . . . . . . . 82
iv
LIST OF FIGURES
7.9 A simulated avalanche, drift and the diffusion of secondary electrons(blue) and ions (yellow). . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.10 Variation of electron transmission and efficiencies as a function of driftfield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.11 Variation of electron transmission and efficiencies a function of induc-tion field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.12 Variation of electron transmission and efficiencies a function of GEMvoltage (For different pitch). . . . . . . . . . . . . . . . . . . . . . . . 86
7.13 Variation of IBF a function of drift and induction field. . . . . . . . . 877.14 Variation of IBF as a function of GEM voltage (For different pitch). . 887.15 Variation of Gain as a function of GEM Volatge. . . . . . . . . . . . . 89
v
List of Tables6.1 Comparison of theoretical calculation and neBEM field values. . . . . 72
vi
Chapter 1
IntroductionThe history of the nuclear and elementary particle physics has seen the development
of many different types of detectors to detect and quantitatively measure various
types of radiations. In order to detect a radiation we convert it’s energy to a known
measurable form. For example when a charged particle or radiation passes through
gas it loses it’s energy which can be collected in a electrical signal by applying electric
field.
Particles and radiation can be detected only through their interactions with mat-
ter. There are specific interactions for charged particles which are different from those
of neutral particles, e.g. of photons. One can say that every interaction process can
be used as a basis for a detector concept. The variety of these processes is quite rich
and, as a consequence, a large number of detection devices for particles and radia-
tion exist.(Fig-1.1) In addition, for one and the same particle, different interaction
processes at different energies may be relevant. Calculations for gas-based detectors
have 3 main components: ionisation, field calculation and charged particle transport.
A Time Projection Chamber (TPC), is an advanced particle detector for detect-
ing charged particles. It is capable of 3D tracking. The momentum measurements
and particle identification can be achieved through energy loss measurements, if it is
placed in a solenoidal magnetic field. A TPC usually consists of a gas filled cylindrical
drift volume. An uniform electric field is applied over the drift volume, usually with
the anodes at the two ends of the chamber and the cathode in the middle, dividing
the chamber in two halves. The homogeneity of the electric field is provided by a
1
1 Introduction
Figure 1.1: Various types of detection techniques
field cage that consists of equidistant ring of electrodes placed along the TPC. The
ring-electrodes are powered with constant potential differences between the cathode
and anode plane. When a charged particle crosses the gas, it produces ionization
along its trajectory. Driven by the electric field, the electrons separate from the gas
ions and drift towards the read-out plane. This means that the electric field shifts an
image of the trajectory to the read-out - in the best case without any deformations.
Directly in front of the read-out plane, the electrons are multiplied in the amplifi-
cation structures. The amplification is necessary because an initial particle creates
only about 100 electron-ion pairs per centimeter which are too small to produce suf-
ficiently strong signals on the read-out plane. Finally, a signal density distribution is
measured which represents a projection of the trajectory onto the read-out plane. X
and Y coordinates are obtained from the detection plane and the Z coordinate from
the drift time. A computer algorithm then reconstructs the trajectory from the signal
2
1 Introduction
distributions on the pads and corresponding drift times. A schematic diagram of the
TPC and its operating principle are shown in Figure below.
The conventional read-out system in the TPCs are the MWPCs.. But the ion
feedback problem restricts the use of the MWPC in high rate experiments. The
ions created in the amplification process, drift back into the TPC volume, create
local perturbations in the electric field and, thus, affect the drift behavior of the
electrons from a later track. This can be solved by using an additional plane of
gating wires which has to be activated through some trigger mechanism. Compared
to a typical MWPC , a typical triple GEM detector represents roughly 3 times the
material budget. In many other aspects GEM detectors exceed the performance
of wire chambers. Most notably, GEM detectors feature probably the highest rate
capability of any gaseous detector. GEM s are also much more resilient to aging,
particularly in Ar/CO2 gas mixtures. What makes GEM detectors attractive from a
practical point of view are a relatively low cost of production and operation, and the
fact that they are essentially maintenance free.
There are some intrinsic differences between these two gas detectors. In a GEM
detector, the GEM foil screens the movement of ions (above the GEM ) from the
3
1 Introduction
readout elements (below), thereby eliminating the characteristic ion tails in the signals
of MWPC s. In wire chambers, ions generated in the avalanche remain in the gas
for tens or hundreds of microseconds, reducing the field strength around the anode
wires (and therefore the gas gain). The ions generated in a GEM avalanche leave the
hole within ∼ 100 ns, which explains the orders of magnitude higher rate capability
of GEMs.
Wire chambers also suffer from aging. The mechanisms and failure modes depend
on the radiation environment and gas mixture. The avalanche plasma tends to intro-
duce free radicals in the gas, and deposit various sorts of chemical compounds on the
wire surface. The results are often non-uniformity of gain over the sensitive area and
broken wires. GEMs have proven to be much less prone to aging, and their limits of
integrated charge still need to be found.[10]
We have described here the principles of operation of the gas ionization chamber
focusing on the most relevant mechanisms involved in the detection process. This
knowledge is necessary to understand, simulate and optimize the performance of the
proposed or any existing detection system. The simulation procedure is now-a-days
widely utilize in designing such detectors. Starting with the basic principle of apper-
tain of gas detectors, we first discuss the different regions of operation of gas detec-
tors. Following which, the production of primary and secondary electrons, transport
of charged particle in gases, the electron multiplication process, the required charac-
teristics of the gas mixture have been described. Then we illustrate the basic features
of some gas detectors. As a final point we give a brief idea about the simulation
and present some calculation utilizing various computer programmes. The program
GARFIELD, written by Rob Veenhof, is the most widely used tool for drift chamber
simulation. For calculation of the primary ionization of fast particles in gases, the
program HEED, written by Igor Smirnov, is widely used. A very popular program
4
1 Introduction
for calculation of electron transport properties in different gas mixtures is the pro-
gram MAGBOLTZ, written by Steve Biagi. MAGBOLTZ and HEED are directly
interfaced to GARFIELD, which therefore allows a complete simulation of the drift
chamber processes, from the passage of the charged particle to the detector output
signal.
5
Chapter 2
Basic Principles of interaction of ra-diation with matterA gas ionization chamber consists of a closed volume filled with the adequate gas
within an electric field between an anode and cathode plate. When a particle enters
into the volume, it interacts with the gas and ionize the atoms of the gas. If the
electric field is intense enough, the electrons are highly accelerated and they further
ionize the gas. Under these conditions, the number of electrons grows rapidly forming
what is known as avalanche. The created electrons and ions drift towards the anode
and cathode. The motion of electrons and ions in the gas gives rise to an electric
signal on the electrodes which can be detected by an external circuit. For a given
gas, the magnitude of the induced signals depend mainly on the field strength and
one can distinguishes various modes and regions of operation.
2.1 Interaction of radiation and matter
The effects of a particle traveling through matter depend on its nature and the en-
vironment encountered. The electric charge of the particle or its absence determines
the type of interaction or behaviour.
A charged particle passing through the electronic clouds of atoms would continu-
ously tear away some of their electrons, a phenomenon called ”ionization” . Neutral
particles, gamma or neutron do not interact in a progressive manner. They set in
motion of charged particles : a gamma would knock on an atomic electron, a neu-
tron a proton or a light nucleus. These secondaries particles will in turn ionize the
6
2 Basic Principles of interaction of radiation with matter
atoms. Ionised atoms generally reorganize themselves by emitting photons, among
them characteristics X-rays.
2.1.1 Interaction of charged particles in a gas
An electrical charge (just like gravitational mass) gives a particle the ability to act
over large distances, allowing it to remove electrons from atoms in the material it
passes through. This phenomenon is known as ionization, and the neutral atoms that
have lost electrons have become ions. The greater the electrical charge on a particle,
the greater its ionizing capability. An alpha or beta particle has an energy some
hundreds of thousands of times larger than the few electron-volts needed to ionize
most atoms. It would take some 30 eV to remove an electron from a gaseous atom,
and far less to remove one from a crystalline structure. The silicon crystals used in
certain detectors (and microchips), for instance, only need 3 eV for an electron to be
liberated. An alpha particle with an energy of 3 MeV, therefore, can either ionize
some 100,000 gaseous atoms or close to a million silicon crystals atoms.
From Bethe-Bloch formula (2.1) it is clear that the energy loss due to ionization
is proportional both to the particles mass and to the square of its charge. It also
changes dramatically with speed: when the particle is slower, it spends more time
in the atom and has a higher chance of interacting with it while it passes through.
As a result, ionization becomes particularly intense near the end of the trajectory,
when the particles have lost most of their energy. This property is frequently used in
therapy; the irradiation of cancerous tumors relies on curving trajectories so that the
radioactive particles target the malignant cells.Liberating many electrons uses a great
deal of energy, and the particles finally come to a halt. The greater the ionization
capability, the shorter the distance these particles can cover. This is especially true
7
2 Basic Principles of interaction of radiation with matter
for the alpha particles, where the length of the trajectory depends on their initial
energy.
Beta particles have winding tracks that nevertheless cannot surpass a certain
maximum length characteristic of the initial energy. From a protection point of
view, a layer of armour that is thicker than this maximum length should achieve the
required levels of safety. Electrically charged particles lose energy progressively as
they travel: by liberating electrons from the atoms that they come across. They
gradually slow down and then come to a complete stop. Alpha particles are small
nuclei whose energy is not high enough (with a few exceptions) to induce a nuclear
reaction. They interact essentially with the medium they pass through because of
their electric charge. They are strongly ionizing and easy to stop, but are much more
dangerous than beta rays in case of ingestion or inhalation.
Nuclear fission that takes place in nuclear reactors produces also charged nuclei
(known as fission products) that inherit most of the liberated energy. This energy will
slowly be dissipated through interactions with atoms along a very short trajectory.
Heavier than alpha particles, fission products have a proportionally greater electric
charge. They are extremely ionizing. Each of them take away an energy some 25
times higher than the 4 MeV given to an alpha particle. This energy is converted
into heat, which is at the origin of the electricity produced by nuclear reactors.
Electrons or positrons also interact with the medium through their electric charge,
which is far smaller than that of a nucleus. While alpha particles travel in short,
straight lines, beta particles (some 8,000 times lighter) have long, unpredictable,
chaotic trajectories with numerous abrupt changes of direction.
As a result of their low mass, beta particles are faster than their alpha cousins. An
electron with an energy of 20 keV, considered as low, can travel at 82,000 kilometres
per second, while an electron with 1 MeV (1000 keV) of energy can go as high as
8
2 Basic Principles of interaction of radiation with matter
282,000 kilometres per second almost the 300,000 kilometres per second that light
can travel in vacuum. When electrons travel through a transparent medium, such as
glass or water, they can go faster than the speed of light for that specific medium.
As a result, they emit a characteristic type of light known as Cherenkov radiation.
These electrons are known as relativistic. A complete description of their be-
haviour requires knowledge of Einsteins theory of relativity. Highly relativistic elec-
trons and positrons which pass near a nucleus experience another phenomenon known
as Bremsstrahlung or braking radiation. Under the influence of the nucleus intense
electric field, these particles emit a gamma or X- photon which takes away some of
their energy. As a result, the electron slows down. These Bremsstrahlung photons
that accompany beta radiation need to be considered in radioprotection.
2.1.2 Interaction of neutral particles in a gas
One can say this as energy transfer by proxy. The neutral particles that are of interest
in the field of radioactivity are gamma photons, neutrons produced in the heart of
nuclear reactors, and neutrinos. These neutral particles lose their energy in a given
medium by causing electrically charged particles to move. Gamma photons essentially
interact with electrical charges. Neutrons are nucleons that do not feel electric charges
but can nevertheless be captured by nuclei and trigger nuclear reactions such as
nuclear fission.
Neutrinos are so weakly interacting that they are very difficult to detect. Practi-
cally invisible, they have a negligible effect on matter and it is therefore so easy to
protect ourselves from them that we can ignore their existence. A gamma photon has
the same nature as visible light. As long as the photon does not undergo a collision
with an electron or a nucleus, it does not lose energy in the medium it passes through.
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2 Basic Principles of interaction of radiation with matter
It also causes no damage. When it interacts, however - generally with an electron
or an atom - it transfers most or all of its energy to the medium via three processes
which either eject an electron away from its atom (photoelectric and Compton effects)
or create a new electron and positron (pair production).
Once these electrons start moving, they lose their energy and slow down like any
other charged particle. Somehow, a gamma photon outsources to these particles set
in motion the task of transferring its energy to the medium. Not very localized (and
therefore highly diffuse in space), it is impossible to predict where the transfer of
energy will take place contrarily to the case of alpha or beta particles. Still more
unpredictable is the path of a neutron, the way in which it slows down and loses its
energy. A neutron completely ignores the electrons it passes, interacting only with
nuclear matter. It transfer its energy by knocking nuclei and setting them in motion.
It can also be captured by nuclei, often inducing nuclear reactions. After capturing
one neutron, stable nuclei may become radioactive.
2.2 Ionisation Detectors
Ionization detectors were the first electrical devices developed for radiation detection.
These instruments are based on the direct collection of the ionization electrons and
ions produced in a gas by passing radiation. During the first half of the 20th century,
three basic types of detector were developed: the ionization chamber, the proportional
counter and the Geiger-Muller counter. During the late 1960’s, a renewed interest
in gas ionization instruments was stimulated in the particle physics domain by the
invention of the multi-wire proportional chamber. These devices were capable of
localizing particle trajectories to less than a millimeter and were quickly adopted in
high-energy experiments. Stimulated by this success, the following years saw the
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2 Basic Principles of interaction of radiation with matter
development of the drift chamber and, somewhat later, the time projection chamber.
Because of their higher density, attention has also been focused on the use of liquids
as an ionizing medium. The physics of ionization and transport in liquids is not as
well understood as in gases, but much progress in this domain has been made and
development is continuing in this area.
2.2.1 Gaseous ionisation Detectors
Because of the greater mobility of electrons and ions, a gas is the obvious medium to
use for the collection of ionization from radiation. Many ionization phenomena arise
in gases and over the years these have been studied and exploited in the detectors
we will describe below. The three original gas devices, i.e., the ionization chamber,
the proportional counter and the Geiger-Muller counter, serve as a good illustration
of the application of gas ionization phenomena in this class of instruments. These
detectors are actually the same device working under different operating parameters,
exploiting different phenomena. Consider a chamber (Fig-2.1) with two electrodes
and voltage applied across them. The chamber is filled with a suitable gas, usually a
noble gas such as argon. If radiation now penetrates the chamber, a certain number
of electron-ion pairs will be created, either directly, if the radiation is a charged
particle, or indirectly through secondary reactions if the radiation is neutral. The
mean number of pairs created is proportional to the energy deposited in the counter.
Under the action of the electric field, the electrons will be accelerated towards the
anode and the ions toward the cathode where they are collected.
The current signal observed, however, depends on the field intensity. This is
illustrated in Fig.2.2 which plots the total charge collected as a function of V. At zero
voltage, of course, no charge is collected as the ion-electron pairs recombine under
their own electrical attraction. As the voltage is raised, however, the recombination
11
2 Basic Principles of interaction of radiation with matter
Figure 2.1: Schematic setup for a gaseous ionisation detector.
forces. are overcome and the current begins to increase as more and more of the
electron-ion pairs are collected before they can recombine. At some point, of course,
all created pairs will be collected and further increases in voltage show no effect.
This corresponds to the first flat region in Fig.2.2 . A detector working in this
region (II) is called an ionization chamber since it collects the ionization produced
directly by the passing radiation. The signal current, of course, is very small and
must usually be measured with an electrometer. Ionization chambers are generally
used for measuring gamma ray exposure and as monitoring instruments for large
fluxes of radiation. If we now increase the voltage beyond region II we find that the
current increases again with the voltage. At this point, the electric field is strong
enough to accelerate freed electrons to an energy where they are also capable of
ionizing gas molecules in the chamber. The electrons liberated in these secondary
ionisation then accelerate to produce still more ionization and so on. This results
12
2 Basic Principles of interaction of radiation with matter
Figure 2.2: I-V characteristics of Gas detectors.
in an ionization avalanche or cascade. Since the electric field is strongest near the
anode, as in most of the configurations, this avalanche occurs very quickly and almost
entirely within a few milimeter from the anode. The number of electron-ion pairs in
the avalanche, however, is directly proportional to the number of primary electrons.
What results then is a proportional amplification of the current, with a multiplication
factor depending on the working voltage V. This factor can be as high as 106 so that
the output signal is much larger than that from an ionization chamber, but still
in proportion to the original ionization produced in the detector. This region of
proportional multiplication extends up to point (III) and a detector operating in this
domain is known as a proportional chamber. It is the important mode for many
sophisticated gas devices to be described later.
If the voltage is now increased beyond point (III), the total amount of ionization
created through multiplication becomes sufficiently large that the space charge created
13
2 Basic Principles of interaction of radiation with matter
distorts the electric field about the anode. Proportionality thus begins to be lost. This
is known as the region of limited proportionality. Increasing V still higher, the energy
becomes so large that a discharge occurs in the gas. What happens physically is that
instead of a single, localized avalanche at some point along the anode wire (as in
a proportional counter), a chain reaction of many avalanches spread out along the
entire length of the anode is triggered. These secondary avalanches are caused by
photons emitted by deexciting molecules which travel to other parts of the counter to
cause further ionizing events. The output current thus becomes completely saturated,
always giving the same amplitude regardless of the energy of the initial event. In order
to stop the discharge, a quenching gas must be present in the medium to absorb the
photons and drain their energy into other channels. Detectors working in this voltage
region are called Geiger-Muller or breakdown counters. The Geiger voltage region, in
fact, is characterized by a plateau over which the count rate varies little. The width
of the plateau depends on the efficacy of the quencher in the gas. In general, the
working voltage of a Geiger counter is chosen to be in the middle of the plateau in
order to minimize any variations due to voltage drift.
Finally, now, if the voltage is increased still further a continuous breakdown oc-
curs with or without radiation. This region, of course, is to be avoided to prevent
damage to the counter. In this illustration, we thus see how phenomena such as gas
multiplication and discharge, in addition to gas ionization, can be used for radiation
detection.
2.2.2 Choice of Gas
Two of the most important parameters for the detector performance are spatial res-
olution and efficiency. The resolution is directly influenced by the diffusion and the
number of ionisations in the gas, whereas the efficiency only depends on the number
14
2 Basic Principles of interaction of radiation with matter
of ionisations. One contribution to the resolution is the diffusion D(z) that goes with
√z. Subsequently, a good resolution requires low diffusion. Many ionisations fur-
ther improves the resolution. An additional requirement is the drift velocity, that is
specifically important in high rate experiments; pile-up can be reduced by increasing
the drift velocity.
The basic component of a gas mixture is the carrier gas. Because gas multiplication
is critically dependent on the migration of free electrons rather than much slower pos-
itive ions, the fill gas in proportional counters must be chosen from those species that
do not exhibit an appreciable electron attachment coefficient. Because air is not one
of these, proportional counters must be designed with the provision to maintain the
purity of the gas. The influence of electronegativity impurities is most pronounced in
large volume counters and for small values of electron drift velocity. In CO2, a gas
with relatively slow electron drift, it has been shown that an Oxygen concentration
of 0.1% results in the loss of approximately 10% of free electrons per centimeter of
travel.
A non-flammable, eco-friendly gas mixture is often a pre-requisite for safety. Hence
the inert gas like Helium, Argon, Neon or Krypton can be used. These gases have
the advantage that multiplication occurs already at lower fields than in other gases.
But these gases have the property that they rather become excited than ionized, since
their ionization energies are rather high. Since low energy transfers are more prob-
able, the excitation reaction generally dominate. As a consequence of de-excitation,
high energetic photons are created. Under proper circumstances, these photon could
create additional ionization elsewhere in the fill gas through photoelectric interac-
tion with less tightly bound electron shell or capable of ionizing the cathode, causing
further avalanches. These can lead to a loss of proportionality and spurious pulses.
Furthermore, they increase the possible dead time effects, reduce the spatial resolu-
15
2 Basic Principles of interaction of radiation with matter
tion in position sensing detector.
To avoid this, quenchers are used. Quenchers are mainly organic (polyatomic) gases,
which have many non-radiative rotational and vibrational levels by which photos can
be absorbed, e.g. Methane or Isobutane. Quenchers therefore help to increase the
operation stability. A mixture of 90% Argon and 10% Methane is probably the most
general purpose proportional gas. The ageing effect is major problem. To clean the
chamber, in general, other gases, like CF4, CF3Br or alcohols like Isopropyl alcohol
are used. Alcohol molecules concentrate near the anode since they have large dipole
moments and will perform charge exchange with the charged ionic molecules. This
will avoid their possible damage. A permanent gas circulation also slows down the
process of ageing.
Even though Oxygen is electronegative, the air can serve as an acceptable propor-
tional gas under special circumstances. If the drift distance of electron is very small
and the electric field in the drift region is high enough, then electrons escape attach-
ments to form small but detectable avalanches. Alternatively, electrons may attach
O2 molecules in the air and drift slowly to near the anode where some are detected
through collision and multiplied by initiating avalanches.
2.3 Energy loss
Fast charged particles, while traveling through matter, lose kinetic energy through
the following mechanism:
1 Coulomb Scattering: Inelastic collisions with atoms; the part of energy is trans-
ferred to excite or ionize the atoms.
2 Bremsstrahlung: Emission of a photon due to the deflection of the particle in
the Coulomb field of a nucleus or an electron.
16
2 Basic Principles of interaction of radiation with matter
The heavy charged particles lose energy essentially by inelastic collision, whereas
for the electrons, the bremsstrahlung starts to dominate above 10-100 MeV. For
identification of the particles, we need to know the energy loss per unit distance of a
moving charged particle in a medium. The mean energy loss of a particle traversing a
gas volume can be calculated using the Bethe-Bloch equation. The basic assumption
of Bethe-Bloch formula are:- The heavy particles interact with atomic electrons which
are assumed to be free and initially at rest. It only moves very slightly during the
interaction with the heavy particle. After the collision, we assume the incident particle
to be essentially undeviated from its original path because of its much larger mass.
The following equation gives the mean energy loss of a traversing particle along a
distance
−dEdx
= 2πNanerec2ρ
Zz2
Aβinc
[ln
(2meγ
2v2Wmax
I2
)− 2β2
inc
](2.1)
where E : energy of the traversing particle ; c: velocity of light
re : classical electron radius ; me : electron mass
Na : Avogadros number ; I: mean excitation potential
Z: atomic number of absorbing material ; A: atomic weight of absorbing material
ρ : density of absorbing material ; z: charge of incident particle
βinc : vc
of the incident particle & γ : 1√1−β2
inc
Wmax : maximum energy transfer in a single collision
Bethe-Bloch formula for the energy loss of a charged particle in a medium is well
known but gives only the mean integrated energy loss. The energy loss is however a
statistical process and will, therefore, fluctuate from event to event. To simulate the
true signal in a proportional chamber, it is necessary to use a much more detailed
model that gives the distribution of the individual ionization along the track and their
17
2 Basic Principles of interaction of radiation with matter
energies. The photo absorption ionization (PAI)model is one of the such models. It
was developed using a semiclassical approach. In the simulation programme HEED,
this PAI model is used for the calculation of the ionization and cluster formation.
An effective description of the ionization left by the particle along its trajectory is
provided by a probability distribution of the number electrons liberated directly or
indirectly with each primary encounter. It is known as cluster size distribution. For
most of the primary ionizing collisions, the energy transfers are small. As a result
the secondary ionization is produced close to the primary collisions and the total
ionization along the particle tracks appears in clusters.
2.4 Different types of ionisation mechanism
The main interactions of charged particles with matter are ionisation and excitation.
For relativistic particles, bremsstrahlung energy losses must also be considered. Neu-
tral particles must produce charged particles in an interaction that are then detected
via their characteristic interaction processes. In the case of photons, these processes
are the photoelectric effect, Compton scattering and pair production of electrons.
The electrons produced in these photon interactions can be observed through their
ionisation in the sensitive volume of the detector.
When the incident particle passes through matter, it transfers a part of its energy
to atoms through collisions with them. This energy is dissipated in matter by emis-
sion of a series of electrons and photons which ionize other atoms and so on. Different
processes, that can take place, are described below with symbols :-
Primary processes :-
excitation:X → X∗
18
2 Basic Principles of interaction of radiation with matter
ionization: X → X+ + e−
dissociation:X → Y ∗ + Z∗
elastic collision: X −→ X
Now using these many types of secondary process can also occur as following
secondary processes :-
non-radiative transitions: X∗ + Y −→ X + Y ∗
radiative transitions: X∗ −→ X + hν
Penning effect: X∗ + Y −→ X + Y + + e−
charge exchange: X+ + Y −→ X + Y +
electron capture: X + e− −→ X− + hν
recombination: X+ + e− −→ X + hν
secondary ionization: e− +X −→ X+ + 2e−
Some of the processes are described in detail later.
Figure 2.3: Ionisation mechanisms considering Bohr’s model of an atom.
19
Chapter 3
Energy loss of Charged Particles3.1 Bethe-Bloch formula
Let us discuss an approximate derivation. In this way, the physical meaning of the
terms appearing in the formula and their behavior as a function of incoming velocity
become more evident. We restrict ourselves to cases where only a small fraction of
the incoming kinetic energy is transferred to atomic electrons, so that the incoming
particle trajectory is not deviated. Now, we introduce the impact parameter b de-
scribing how close the collision is : b is the minimal distance of the incoming particle
to the target electron. In general, large values of b correspond to the so-called distant
collisions, conversely small values to close collisions. Both kinds of collisions are im-
portant for determining the average energy-loss [Eq. 2.1], the energy straggling (i.e.,
the energy-loss distribution) and the most probable energy-loss. When a particle of
charge z1e,mass M and velocity v interacts with a particle in the material with mass
m, charge z2e and we assume that, to a first approximation, the material particle will
emerge only after the particle passage so that we can consider the electron essentially
at rest throughout the interaction. We restrict ourselves to cases where only small
momentum transfers are involved, so that the trajectory of the incident particle is not
appreciably altered and the material particle only has a small recoil. The trajectory
of the incident particle defines the axis of a cylinder as shown in fig:- We cosider the
interaction with a particle in the cylindrical shell a distance b from the axis. The
distance b is reffered to as the impact parameter for the interaction.
The moving charge creates an electric and magnetic field at the location of the ma-
20
3 Energy loss of Charged Particles
Figure 3.1: Incoming fast particle of charge ze scattered by an atomic electron almostat rest: for small energy transfer, the particle trajectory is not deflected
terial particle. Since the material particle is assumed to have only a small velocity,
the magnetic interaction is not important. By symmetry the net force acting on the
material particle is perpendicular to the cylinder.
The transverse electric field is
E⊥ =z1eb
r3(3.1)
in the rest frame of the incident particle. The electric field obsesrved in the LAB
changes with time. Suppose that the incident particle reaches its pointy of closest
approaxh at t = 0. At time t the transeverse electric field in LAB frame is given by
[18]
E⊥ =γz1eb
(b2 + γ2v2t2)3/2(3.2)
The momentum acquired by the bound particle is
4p =
∫Fdt =
∫ ∞−∞
(z2e)γz1eb
(b2 + γ2v2t2)3/2dt =
2z1z2e2
vb(3.3)
The incident particle will have collisions with both the nuclei and the electrons
of the atoms. Since the bound particle is assumed to have only a small velocity,
21
3 Energy loss of Charged Particles
Figure 3.2: An incoming fast particle of charge ze interacts with electrons at impactparameter between b and b+ db
the energy transefer can be written (caution same symbol used for electric field and
energy, clear from the context)
4E =(4p)2
2m=
2z21z22e
4
b2v2m(3.4)
Inverse relation with the impact parameters says that most of the energy transefer
is due to close collisions. We have m = me and z2 = 1 for electrons and m = Amp
and z2 = Z for nuclei. With Z electrons in an atom and A ≈ 2Z,
4E(electrons)
4E(nucleus)=
Z
me
(Z2
2Zmp
)−1≈ 4000 (3.5)
so we see that the atomic electrons are responsible for most of the energy loss. We
will let m = me for rest of the disscusion.
Now let us calculate the total energy lost by the incident particle per unit length
in the material. We have just seen that most of the energy loss is due to interactions
with the atomic electrons. There are ne × 2πb dbdx electrons in the cylindrical shell
of Figure(3.2), where
ne = z2na = z2Naρ
A(3.6)
22
3 Energy loss of Charged Particles
is the number of the electrons per unit volume. Summing over the total energy
transfer in each b interval, the total energy loss per unit length is
−dEdx
= 2πne
(2z21e
4
mv2
)∫ bmax
bmin
db
b=
4πnez21e
4
mv2lnbmaxbmin
(3.7)
The limiting values of the impact parameter are determined by the range of validity
of the various assumptions that were made in deriving last equation. We have assumed
that the interaction takes place between the electric field of the incident particle and
a free electron. However, the electron is actually bound to an atom. The interaction
may be considered to be with a free electron only if the collision time is too short
compared to the characterstic orbital period of electrons in the atom. Examination
of Eq. 3.2 shows that the transeverse electric field in the LAB is very small except
near t = 0. The full width at half maximum of the electric field E(t) distribution is
bvγ
times a comnstant of order 1, so we take [18]
tcoll 'b
vγ(3.8)
An upper limit for the impact parameter then is
bmax 'γv
ω=γvh
I(3.9)
where ω is a characterstic orbital frequency and I = hω is the mean excitation
energy. The lower limit bmin is evaluated considering the extent to which the classical
treatment can be employed. In the framework of the classical approach, the wave
characteristics of particles are neglected. This assumption is valid as long as the
impact parameter is larger than the de Broglie wavelength of the electron in the
center-of-mass system (CoMS) of the interaction. For instance, we can assume
bmin 'h
2PeCM(3.10)
23
3 Energy loss of Charged Particles
where PeCM is the electron momentum in the CoMS. Because the electron mass
is much smaller than the mass of the incoming heavy-particle, the CoMS is approxi-
mately associated with the incoming particle and conversely the electron velocity in
the CoMS is opposite and almost equal in absolute value to that of the incoming
particle, v. Thus, we have that
|PeCM | ' mγv = mγβc (3.11)
and substituting in equation-3.7 we obtain:
−dEdx
=4πnez
21e
4
mv2ln
[(vhγ
I
)(2mγβc
h
)]=
2πnz2e4
mv2ln
(2mγ2v2
I
)2
(3.12)
Finally, using the value of the maximum energy transfer Wm = 2mv2γ2 from two
body scattering we get:
−dEdx
=2πnz2e4
mv2ln
[2mγ2v2Wm
I2
](3.13)
This equation is equivalent to the energy-loss formula except for the terms 2β2, −δ
due to the density-effect and the shell correction term (−U). When the first of these
latter terms is added, we obtain the expression (termed Bethe relativistic formula)
−dEdx
=4πnz2e4
mv2
(ln
[2mγ2v2
I
]− β2
)(3.14)
derived in the quantum treatment of energy loss by collisions of a heavy, spin-0 inci-
dent particle.
Furthermore, it has to be noted that spin plays an important role when the transferred
energy is almost equal to the incoming energy (this occurs with very limited statistical
probability). At low particle velocity [the energy-loss process due to Coulomb inter-
actions on nuclei (termed nuclear energy-loss and resulting in the nuclear stopping
24
3 Energy loss of Charged Particles
power ) cannot be neglected], additional corrections are added: for instance, correc-
tions accounting for the Barkas effect [denotes the observed difference in ranges of
positive and negative particles in emulsion] and the Bloch correction[derives from the
Bloch quantum-mechanical approach in which he did not assume, unlike Bethe, that
it is valid to consider the electrons to be represented by plane waves in the center-
of-momentum reference frame].
25
Chapter 4
Transport of charged particle in GasesA charged particle under the effect of an electric field is accelerated along the field
lines.Here the drift velocity of electrons & ions and the diffusion of electrons are
described.To model the slowing down of the drifting particles by the gas molecules,
a friction force, proportional to the velocity, has been introduced. The behaviour
of the drift chamber is crucially dependent on the drift of the electrons and ions
that are created by the particles measured or in the avalanches at the electrodes. In
addition to the electric drift field, there is often a magnetic field, which is necessary
for measuring particle momentum. Obviously we have to understand how the drift
velocity vector in electric and magnetic fields depends on the properties of the gas
molecules, including their density and temperature.
Differences in behaviour of ions and electrons : As might be expected, elec-
trons usually have much higher drift velocities and diffusion coefficients (by orders of
magnitude) than ions under given conditions in a given gas. Because of their small
mass, electrons are accelerated rapidly by an electric field, and they lose little energy
in elastic collisions with molecules (a fraction of the order of me/M , where me and
M are the electronic and molecular masses, respectively.) Therefore, electrons can
acquire kinetic energy from an electric field faster than ions, and they can store this
energy between collisions to a much greater degree until they reach energies at which
inelastic collisions become important. Even with only a weak electric field imposed
on the gas through which the electrons are moving, the average electronic energy may
be far in excess of the thermal value associated with the gas molecules. Furthermore,
26
4 Transport of charged particle in Gases
the electronic energy distribution is not close to Maxwellian except at extremely low
values of E/N.
Other differences between electrons and ions develop in connection with their
collision cross sections. Electronic excitation of atoms and molecules is frequently an
important factor in electron collisions even for impact energies of less than 10 eV, and
in molecular gases the onset of vibrational and rotational excitation occurs at energies
far below 1 eV. These energies are often attained by electrons in situations of common
interest. The laboratory-frame thresholds for the corresponding modes of excitation
by ions are higher than those for electrons, and the excitation cross sections peak
at energies considerably above these thresholds. Therefore, ions have insufficient
energy to produce much excitation under the usual gas kinetic conditions. These
considerations tend to make the analysis of electronic motion in gases more difficult
than the analysis of ionic motion. A compensating factor is operative, however,
because of the relatively small mass of the electron. Since me/M << 1 in any gas,
it is possible to make approximations in the analysis of electronic motion that are
not valid in the ionic case. These approximations greatly simplify the mathematics
and make it possible to calculate accurately the velocity distribution and transport
properties of electrons in many gases at high E/N. This is not so with ions.
On the experimental side certain other important differences appear. Electrons
may be produced much more simply than ions by thermionic emission from filaments,
by photoemission from surfaces, or by beta decay of radioactive isotopes. Ionic pro-
duction usually requires the use of much more elaborate apparatus: electron bom-
bardment or photoionization ion sources or an electrical discharge. Furthermore, in
an electron-swarm experiment the electronic component of the charge carriers may
be easily separated from any ionic component that may be present and no mass anal-
ysis is required to interpret the data. In ionic drift and diffusion experiments, on
27
4 Transport of charged particle in Gases
the other hand, mass analysis of the ions is usually essential if unambiguous results
are to be obtained. This requirement involves a great complication of the apparatus.
On the other hand, electron swarm experiments are usually more sensitive than ion
experiments to electric field non uniformities, contact potentials, and magnetic fields.
A final difference between electron and ion experiments relates to the effects of
impurities in the gas being studied. Molecular impurities in an atomic gas can hold
the average electronic energy well below the level that would be attained in the pure
gas because electrons can lose large fractions of their energy by exciting the rotational
and vibrational levels of the molecules. The electronic velocity distribution can be
seriously altered in the process. In ionic experiments, however, impurities have little
effect on the average ionic energy and velocity distribution. The complication that
may develop instead is the production of impurity ions by the reaction of the ions of
the main gas with impurity molecules. This is frequently a matter of serious concern.
4.1 Drift of Charged Particle
The motion of charged particles under the influence of electric and magnetic fields
obey the following equation of motion :
md~u
dt= q ~E + q(~u× ~B)− f~u (4.1)
where m and q are the mass and electric charge of the particle, u is its velocity vector,
and f describes a frictional force proportional to u that is caused by the interaction
of the particle with the gas. This equation describes the drift at large t to a very good
approximation. This was introduced by P. Langevin, who imagined the force fu as a
stochastic average over the random collisions of the drifting particle. The ratio m/f
has the dimension of a characteristic time, and we define τ = m/f . For very large
time t >> τ we will get a steady state solution withd~u
dt= 0. So we get the linear
28
4 Transport of charged particle in Gases
differential equation for the drift velocity vector as
1
τ~u− q
m[~u× ~B] =
q
m~E (4.2)
In order to solve for ~u, we write (q/m)Bx = ωx etc., (q/m)Ex = εx etc., and
express the above equation in the form of the matrix equation
Mu = ε (4.3)
where
M =
1/τ −ωz ωyωz 1/τ −ωx−ωy ωx 1/τ
After inverting the matrix we get the solution for the drift velocity as
~u = M−1~ε =
1 + ω2xτ
2 ωzτ + ωxωyτ2 −ωyτ + ωxωzτ
2
−ωzτ + ωxωyτ2 1 + ω2
yτ2 ωxτ + ωyωzτ
2
ωyτ + ωxωzτ2 −ωxτ + ωyωzτ
2 1 + ω2zτ
2
× τ
1 + ω2τ 2~ε
(4.4)
where ω2 = ω2x + ω2
y + ω2z = (q/m)2B2 is the square of the cyclotron frequency of the
electron. We can write the same equation as
~u =q
m|E| τ
1 + ω2τ 2
(E + ωτ [E × B] + ω2τ 2(E · B)B
)(4.5)
Here E and B denote the unit vectors in the directions of the fields. The drift direction
is governed by the dimensionless parameter ωτ , where ω is defined as (q/m)|B| and
carries the sign of the charge of the moving particle. For ωτ = 0, ~u is along ~E; in
this case the relation has the simple form
~u =q
mτ ~E = µ~E (4.6)
where µ = qmτ is defined as the scalar mobility, the ratio of drift velocity to electric
field in the absence of magnetic field; µ is proportional to the characteristic time τ
and carries the charge sign of the particle.
29
4 Transport of charged particle in Gases
In the presence of the magnetic field, the mobility is the tensor (q/m)M−1 given
in Eq-4.4 . For large positive values of ωτ and q > 0,~u generally tends to be along ~B;
but if E · B = 0, then large ωτ turn ~u in the direction of E× B, independently of the
sign of q. Both the direction and the magnitude of u are influenced by the magnetic
field. If we distinguish u(ω), the drift velocity in the presence of B, and u(0), the
drift velocity at B = 0 under otherwise identical circumstances, using eq-5 we derive
u2(ω)
u2(0)=
1 + ω2τ 2cos2φ
1 + ω2τ 2(4.7)
where φ is the angle between E and B. This ratio happens to be the same as the one
by which the component of u along E, uE, changes with B:
uE(ω)
uE(0)=
1 + ω2τ 2cos2φ
1 + ω2τ 2(4.8)
4.2 Microscopic Picture
When charges drifting through the gas are scattered on the gas molecules so that
their direction of motion is randomized in each collision. On average, they assume a
constant drift velocity ~u in the direction of the electric field (or, if a magnetic field is
also present, in the direction which is given by both fields). The drift velocity is much
smaller than the instantaneous velocity c between collisions. The gases we deal with
are sufficiently rarefied that the distances travelled by electrons between collisions are
large in comparison with their Compton wavelengths. So our picture is classical and
atomistic.
4.2.1 Drift of Electrons
Consider an electron moving between two successive collisions. Because of its light
mass, the electron scatters isotropically and, immediately after the collision, it has
30
4 Transport of charged particle in Gases
forgotten any preferential direction. Some short time later, in addition to its instanta-
neous and randomly oriented velocity c, the electron has picked up the extra velocity
u equal to its acceleration along the field, multiplied by the average time that has
elapsed since the last collision:
~u =e ~E
mτ (4.9)
This extra velocity appears macroscopically as the drift velocity. In the next en-
counter, the extra energy, on the average, is lost in the collision through recoil or
excitation. Therefore there is a balance between the energy picked up and the colli-
sion losses. On a drift distance x, the number of encounters is n = (x/u)(1/τ), the
time of the drift divided by the average time τ between collisions. If λ denotes the
average fractional energy loss per collision, the energy balance is the following:
x
uτλεE = eEx (4.10)
Here the equilibrium energy εE carries an index E because it does not contain the
part due to the thermal motion of the gas molecules, but only the part taken out of
the electric field.
We have the average time between collisions same as the average time that has
elapsed since the last collision. This is because in a completely random series of
encounters, characterized only by the average rate 1/τ , the differential probability
f(t)dt that the next encounter is a time between t and t + dt away from any given
point t = 0 in time is
f(t)dt = 1/τe−t/τdt (4.11)
independent of the point where the time measurement begins. In the frictional-
motion picture, τ was the ratio of the mass of the drifting particle over the coefficient
of friction. In the microscopic picture, τ is the mean time between the collisions of the
31
4 Transport of charged particle in Gases
drifting particle with the atoms of the gas. For drifting particles with instantaneous
velocity c, the mean time τ between collisions may be expressed in terms of the
cross-section σ and the number density N :
1
τ= Nσc (4.12)
Here c is related to the total energy of the drifting electron by
1
2mc2 = ε = εE +
3
2kT (4.13)
because the total energy is made up of two parts: the energy received from the
electric field and the thermal energy that is appropriate for 3 degrees of freedom
(k = Boltzmanns constant, T = gas temperature). For electron drift in particle
detectors, we usually have εE � (3/2)kT ; we can neglect the thermal motion, and
eq- 4.9,4.10&4.13 combine to give the two equilibrium velocities as follows:
u2 =eE
mNσ
√λ
2(4.14)
c2 =eE
mNσ
√2
λ(4.15)
where ε = (1/2)mc2 ' εE � (3/2)kT
4.3 Drift of Ions
The behavior of ions differs from that of electrons because of their much larger mass
and their chemical reactions.Electrons in an electric field are accelerated more rapidly
than ions, and they lose very little energy when colliding elastically with the gas
atoms. The electron momentum is randomized in the collisions and is therefore lost,
on the average. In electric field strengths that are typical for drift chambers, the
electrons reach random energies far in excess of the energy of the thermal motion,
and quite often they surpass the threshold of inelastic excitation of molecules in the
32
4 Transport of charged particle in Gases
gas. In this case their mobility becomes a function of the energy loss that is associated
with such excitation. Ions in similar fields acquire, on one mean-free path, an amount
of energy that is similar to that acquired by electrons. But a good fraction of this
energy is lost in the next collision, and the ion momentum is not randomized as
much. Therefore, far less field energy is stored in random motion. As a consequence,
the random energy of ions is mostly thermal, and only a small fraction is due to
the field. The effect on the diffusion of ions results in this diffusion being orders of
magnitude smaller than that of electrons in similar fields. The effect on the mobility
is also quite interesting: since the energy scale, over which collision cross-sections
vary significantly, is easily covered by the electron random energies reached under
various operating conditions, we find rapid and sometimes complicated dependencies
of electron mobility on such operating conditions electric and magnetic field strengths
and gas composition being examples. In contrast, the mobility of ions does not vary
as much.
4.4 Inclusion of Magnetic Field
When we consider the influence of a magnetic field on drifting electrons and ions,
the first indication may be provided by the value of the mobility of these charges. In
particle chamber conditions, this is of the order of magnitude of µ ' 104cm2V −1s−1
for electrons whereas for ions the order of magnitude is µ = 1cm2V −1s−1. Now it is
the numerical value of ωτ = (e/m)Bτ that governs the effects of the magnetic field
on the drift velocity vector.
Therefore, the effect of such magnetic fields on ion drift is negligible, and we
concentrate on electrons. This has the advantage that we may assume that the
colliding body scatters isotropically in all directions, owing to its small mass.
33
4 Transport of charged particle in Gases
When the magnetic field is added we can describe the most general case in a
coordinate system in which B is along z, and E has components Ez and Ex. An
electron between collisions moves according to the equation of motion,
md~v
dt= q ~E + q(~v × ~B) (4.16)
so we are getting
vx = εx + ωvy
vy = −ωvx
vz = εz
(4.17)
Electrons have their direction of motion randomized in each collision, and we are
interested in the extra velocity picked up by the electron since the last collision.
Hence we look for the solution of eq-4.16 with v = 0 at t = 0. It is given by
vx = (εx/ω)sinωt
vy = (εy/ω) (cosωt− 1)
vz = εzt
(4.18)
Before we can identify v with the drift velocity u, we must average over t, using (4.11),
the probability distribution of t. This was also done when deriving (4.9), which, being
a linear function of time, required t to be replaced by τ , the mean time since the last
collision. The drift velocity for the present case is given by
ux = 〈vx(t)〉 =εxω
∫ ∞0
1
τe−t/τsin(ωt)dt =
εxτ
1 + ω2τ 2
uy = 〈vy(t)〉 =εxω
∫ ∞0
1
τe−t/τsin(ωt)dt =
εxωτ2
1 + ω2τ 2
uz = 〈vz(t)〉 =εzω
∫ ∞0
1
τe−t/τdt = εzτ
(4.19)
4.5 Diffusion
A charged particle drifting under the influence of external fields scatters off the gas
molecules and does not follow precisely the field lines. A cloud of such particles spread
34
4 Transport of charged particle in Gases
out perpendicular and along the field lines. This process is called diffusion. After a
collision, ions retain their direction of motion to some extend because their mass is
comparable to the mass of the gas molecules. These diffuse little at the typical drift
fields encountered in gas detectors. Electrons, scatter almost isotropically and their
direction of motion is randomized after each collision. As the drifting electrons or
ions are scattered on the gas molecules, their drift velocity deviates from the average
owing to the random nature of the collisions. In the simplest case the deviation is the
same in all directions, and a point-like cloud of electrons which then begins to drift
at time t = 0 from the origin in the z direction will, after some time t, assume the
following Gaussian density distribution:
n(r, t) =
(1
4πDt
)3/2
exp
(−r2
4Dt
)(4.20)
where
r2 = x2 + y2 + (z − ut)2
D is the diffusion constant because n satisfies the continuity equation for the conserved
electron current density Γ (it has both drift and diffusion current):
Γ = nu−D∇n (4.21)
For a detailed explanation of the solution of diffusion equation we will go through
the Fick’s laws and some mathematical physics to solve the differential eqaution. Dif-
fusion is the process by which an uneven concentration of a substance gets gradually
smoothed out spontaneouslye.g., a concentration of a chemical species (like a drop of
ink or dye) in a beaker of water spreads out by itself, even in the absence of stirring.
The microscopic mechanism of diffusion involves a very large number of collisions of
the dye molecules with those of the fluid, which cause the dye molecules to move es-
sentially randomly and disperse throughout the medium, even without any stirring of
35
4 Transport of charged particle in Gases
the fluid. A macroscopic description of the process is based on Ficks Laws, and leads
to a fundamental partial differential equation, the diffusion equation. This equation
serves as a basic model of phenomena that exhibit dissipation, a consequence of the
irreversibility of macroscopic systems in time. The local, instantaneous concentration
n(r, t) of electron swarm satisfies the equation of continuity, which is called Ficks first
law in this context:
∂n
∂t+∇ · Γ = 0 (4.22)
We assume that j is proportional to the local difference in concentrations, i.e., to the
gradient of the concentration itself. Thus
Γ = −D∇n (4.23)
The minus sign on the right-hand side signifies the fact that the diffusion occurs
from a region of higher concentration to a region of lower concentration: That is, the
diffusion current tends to make the concentration uniform. Eliminating Γ, we get the
diffusion equation for the concentration n(r, t):
∂n
∂t= D∇2n (4.24)
This equation is of first order in the time variable, and second order in the spatial
variables. It is a parabolic equation in the standard classification of second-order
partial differential equations. In order to find a unique solution to it, you need
an initial condition that specifies the initial concentration profile n(r, 0), as well as
boundary conditions that specify n(r, t), for all t ≥ 0, at the boundaries of the region
in which the diffusion is taking place. The presence of the first-order time derivative
in the diffusion equation implies that the equation is not invariant under the time
reversal transformation t 7→ −t. Irreversibility is thus built into the description of
the phenomenon.
36
4 Transport of charged particle in Gases
For the fundamental solution in 3 dimensions, we begin with the (eq-4.24) and the
initial and boundary conditions where n(r, t) satisfies natural boundary conditions,
i.e., n(r, t)→ 0 as r →∞ along any direction. We may start with the initial condition
n(r, 0) = δ(r). This means that the diffusing particle starts at the origin at t = 0.
In the context of the diffusion equation for the concentration n(r, t), such an initial
condition represents a point source of unit concentration at the origin. In a space of
infinite extent, we may take the starting point to be the origin of coordinates without
any loss of generality. The solution thus obtained is the fundamental solution (or
Green function) of the diffusion equation. The diffusion equation presents an initial
value problem. Moreover, it is a linear equation in the unknown function n(r, t). It
is therefore well-suited to the application of Laplace transforms (with respect to the
time variable). As far as the spatial variable r is concerned, it is natural to use Fourier
transforms.
Taking the Laplace transform of both sides of the diffusion equation, we get
sn(r, s)− n(r, 0) = D∇2n(r, s)
⇒ (s−D∇2)n(r, s) = δ(r) (4.25)
where n(r, s) =∫∞0dte−stn(r, t), the Laplace transform of n(r, t);
Now expand n(r, s) in a Fourier integral with respect to the spatial variable r, ac-
cording to
n(r, s) =1
(2π)3
∫d3keik·rp(k, s)
p(k, s) =
∫d3ke−ik·rn(r, s)
so using these expressions in the eq-4.25 for n(r, s) and equate the coefficients of the
37
4 Transport of charged particle in Gases
basis vector eik·r in the space of functions of r. We must then have, for each k,
(s+Dk2)p(k, s) = 1
⇒ p(k, s) =1
s+Dk2
We got 1 on the right side by taking the Fourier transform of the delta function of
eq-4.25. We thus obtain a very simple expression for the double transform p(k, s).
The transforms must be inverted to find the n(r, t). It is easier to invert the Laplace
transform first, and then the Fourier transform. The Laplace transform is trivially
inverted, as we have
L−1[1/(s+ a)] = e−at
. Therefore
p(k, t) = e−Dk2t (4.26)
and we will have the solution by taking the inverse Fourier transform of p(k, t)
n(r, t) =1
(2π)3
∫d3keik·re−Dk
2t (4.27)
This 3D integral factors into a product of 3 integrals upon writing r and k in Cartesian
coordinates. Each of the factors is the familiar shifted Gaussian integral. For 1D it
looks like
n(x, t) =1
2π
∫ ∞−∞
dkxeikx·xe−Dk
2xt (4.28)
By completing the square of the exponential part we get the solution
n(x, t) =
(1
4πDt
)1/2
exp
(−x2
4Dt
)(4.29)
For the solution (eq-4.27) we have to multiply 3 of the these and we get eq-4.20 where
we have relaced z by z − ut to consider the drift in z-direction.
From microscopic picture it comes out to be that the diffusion constant
D =l203τ
=c2τ
3=
2ε
3mτ =
2εµ
3e(4.30)
38
4 Transport of charged particle in Gases
We have used the expression for the electron mobility.where l0 = cτ is the mean free
path.
When the diffusing body has thermal energy, ε = (3/2)kT we get the Einstein
formula stated above in (eq-3.4). The diffusion constant D, defined by (3.25), makes
the mean squared deviation of the electrons equal to 2Dt in any one direction from
their centre. The energy determines the diffusion width σx of an electron cloud which,
after starting point-like, has travelled over a distance L:
σ2x = 2Dt =
2DL
µE=
4εL
3eE(4.31)
In drift chambers we therefore require small electron energies at high drift fields in
order to have σ2x as small as possible.
4.6 Electric Anisotropy
After some experimental evidence in 1967 the assumption of the isotropic form of
diffusion constant was considered seriously. Experimentally it was shown that the
value of electron diffusion along the electric field can be quite different from that in
the perpendicular direction. The diffusion of drifting ions is also often found to be
non-isotropic. The value of electron diffusion along the electric field can be quite
different from that in the perpendicular direction. The diffusion of drifting ions is
also often found to be non-isotropic.
When ions collide with the gas molecules, they retain their direction of motion to
some extent because the masses of the two collision partners are similar, and therefore
the instantaneous velocity has a preferential direction along the electric field. This
causes the diffusion to be larger in the drift direction; the mechanism is at work for
ions travelling in high electric fields E[eE/(Nσ) 1 kT ], and we do not treat it here
because it plays no role in the detection of particles. In the case of electrons, there
39
4 Transport of charged particle in Gases
Figure 4.1: Mobility variation inside an electron cloud travelling in the z direction.
is almost no preferential direction for the instantaneous velocity. We will look at
the electron diffusion anisotropy qualitatively. The essential point of the argument is
that the mobility of the electrons assumes different values in the leading edge and in
the centre of the travelling cloud if the collision rate is a function of electron energy.
This change of mobility inside the cloud is equivalent to a change of diffusion in the
longitudinal direction. Instead of the density distribution (4.20) of the diffusing cloud
of electrons, we get in the anisotropic case
n =1√
4πDLt
(1
4πDT t
)exp
[−x
2 + y2
4DT t− (z − ut)2
4DLt
](4.32)
4.7 Magnetic Anisotropy
Let us now consider the effect of a magnetic field B along z. The electric field is in
the x − z plane and we assume there is no electric anisotropy. The magnetic field
causes the electrons to move in helices rather than in straight lines between collisions.
40
4 Transport of charged particle in Gases
Projection onto the x− y plane yields circles with radii
ρ =c
ωsinθ (4.33)
the magnetic field has caused the diffusion along x and y (then perpendicular to the
magnetic field) to be reduced by the factor
DT (ω)
DT (0)=
1
1 + ω2τ 2(4.34)
whereas the longitudinal diffusion is the same as before:
DL(ω) = DL(0) (4.35)
If there is an E field as well as a B field in the gas, the electric and magnetic
anisotropies combine. In the most general case of arbitrary field directions, the dif-
fusion is described by a 3× 3 tensor. The diffusion tensor must be positive definite,
otherwise our cloud would shrink in some direction a thermodynamic impossibility.
4.8 Amplification
In many cases, the signal generated by the primary electrons is not intense enough to
be detected by the read-out electronics. Consequently, the charge of the secondary
electron cloud must be augmented.
4.8.1 Electron multiplication of the gas
The primary charge generated by ionizing radiation in the gas volume is collected on
electrodes by means of an electric field that attracts the the electrons towards the
anode and the ions towards the cathode. But the number of primary electrons, gen-
erated is too small to be detected by the electronics and so has to be increased in the
gas by electron multiplication.At a given gas pressure, the condition is determined
41
4 Transport of charged particle in Gases
mainly by the gas composition and the electric field strength.
• Ionisation by electrons :-
During the migration of these charges, many collisions normally occur with
neutral gas molecules. Because of their low mobility, positive or negative ions
achieve very little average energy between collisions. Free electrons, on the
other hand, are easily accelerated by the applied field and may have significant
kinetic energy when undergoing such a collision. If this energy is greater than
the ionization energy of neutral gas molecule, it is possible for an additional ion
pairs to be created in the collision. Because the average energy of the electron
between collisions increases with increasing electric field, there is a threshold
value of the field above which this secondary ionization will occur. In typical
gases, at atmospheric pressure, the threshold field is of the order of 106 V/m.
The electrons liberated by this secondary ionization process will also be acceler-
ated by the electric field and in turn can cause further ionization. The number
of electrons hence grows with time until all electrons are collected at the anode.
This process is known as electron avalanche. At a given field the mean energy
of the avalanche electrons is higher in hot gases than in cold gases. It would
hence be expected that the largest gains are obtained in noble gases. This is
not true in practice because the multiplication process in these gases are not
stable.
• The role of the photons :-
The cross section for ionization and excitation have roughly the same order of
magnitude at electron energies beyond the inelastic thresholds. Therefore, a
42
4 Transport of charged particle in Gases
comparable number of ionization and excitation occur. In noble gases, the ex-
cited states return to the ground state via the emission of photons. Because ex-
citation mainly concerns outer shell electrons, a direct transition to The ground
state results in the emission of a photon with an energy in the U V range.
De-excitation sometimes involves more than one transition and the energies of
the emitted photons are lower in the IR region. Oppositely, molecular gases
have several excitation levels with non-radiative relaxation modes. They have
a tendency to break into lighter fragments under impact of energetic electrons
IR photons are not sufficiently energetic to impact on the avalanche develop-
ment. But the UV photons can release new electrons from the gas molecules
or from the detector electrodes by the photoelectric effect. The new electrons
initiate secondary avalanches, leading to detector instability and eventually to
detector breakdown. This process are called photon feedback. It is desirable to
stop them early. Molecular gases having absorption bands in the UV range are
suited for this task. They are generally mixed with noble gases to stabilize the
avalanche process.
• Penning effect :-
Penning effect is the ionization of a gas Y by an excited state of a gas X∗.
It has two sources. The first one is when an excited atom (Y ∗ )de-excites by
emitting a photon, which is energetic enough to ionize further atom (Y), which
have a lower ionization energy as the energy carried by the photon. The second
source is when the excited atom has meta-stable states. In this case it cannot
emit a photon. The de-excitation occurs through a collision with a second atom
resulting in the ionization of the later. Beside increasing the primary ionization
43
4 Transport of charged particle in Gases
yield, the Penning effect enhances also the gas gain. The multiplication factor
can not be increased at will. An empirical limit on the maximum charge that
can be tolerated in the avalanche before breakdown was formulated by Raether
and corresponds to an avalanche size of approximately 108 electrons.
4.8.2 The first Townsend coefficient
The first Townsend coefficient (α) is a term used in an ionization process where
there is a possibility of secondary ionization because the primary ionization electrons
gain sufficient energy from the accelerating electric field or from the original ionizing
particle. The coefficient gives the probability of secondary ionization per unit path
length. It is equal to the reciprocal of the mean free path of primary electrons. We
can write the average increase of electrons (dN) over a path dS to be
dN = NαdS (4.36)
The Townsend coefficient α is determined by the excitation and ionization cross sec-
tion of the electrons that have acquired sufficient energy in the field. It depends on
the various transfer mechanism. It also depends on the electric field ~E and increases
with the field because the ionization cross section goes up from the threshold as the
collision energy increases. It also depends on the gas density linearly.
Theoretical consideration of gas amplification [7] in proportional counter has indicated
that the ratio of the first Townsend coefficient to the gas pressure, can be generally
expressed in the form,
α
p= kaS
mexp(−LS1−m ) (4.37)
where S = Ep
is the ratio of the electric field strength to the gas pressure, and
ka,L,m(0 < m < 1) are constant characteristic of the gas. When m = 0, the αp
p
44
4 Transport of charged particle in Gases
formula reduces to the analytical form used by William and Sara (derived from the
models on the behavior of electrons in an electric field) and when m = 1 it reduces
to that assumed by Diethorn ( derived by assuming analytical forms for αp). The
coefficient α as given by William and Sara is
α = pAe−BpE (4.38)
At high electric field values, the Townsend coefficient saturates because its value
approaches the mean free path given by inelastic collision cross section which is con-
stant. In the eq-4.38 A and B are constant which depends on the basic gas composi-
tion.
The amplification factor is given by
G =N
N0
= exp
(∫ S
Smin
α (E(x)) dx
)(4.39)
In case of uniform field, the above equation becomes,
G = eα4x (4.40)
4.8.3 The Second Townsend Coefficient
At high field collisions of electrons with atoms or molecules can cause not only ioni-
sation but also excitation. De-excitation is often followed by photon emission. The
previous considerations are only true as long as photons produced in the course of
the avalanche development are of no importance. These photons, however, will pro-
duce further electrons by the photoelectric effect in the gas or at the counter wall,
which affect the avalanche development. Apart from gas-amplified primary electrons,
secondary avalanches initiated by the photoelectric processes must also be taken into
account.
β =No. of photoelectric effect events
No. of avalanche electrons(4.41)
45
4 Transport of charged particle in Gases
So the gas gain including photoeffect :
Gβ = G+G(Gβ) +G(Gβ)2 +G(Gβ)3 + ... =∞∑k=0
Gk+1βk =G
1− βG(4.42)
Here βG means the average number of electrons released by photons produced in
the first avalanches. In the above equation the first term means no photo effect has
occurred, the second term means one photo effects has occurred and so on. The last
equality holds when β � 1. When βG → 1, continuous discharge independent of
primary ionization has occurred. To prevent this quench gas to absorb UV -photons
are added.
4.9 Electron loss
During the transport of the electrons two main processes decrease the number of
primary electrons. 1) Recombination, 2) Attachment
4.9.1 Recombination
The recombination is the capture of a free electron by a positive ion followed by a
photon emission. For molecular ions, a similar recombination reaction occurs. When
there is no electric field, ion electron pairs will generally recombine under the force
of their electric attraction, emitting a photon in the process. The recombination rate
is proportional to the ions and electron densities.
dn
dt= −bne(t)ni(t) (4.43)
46
4 Transport of charged particle in Gases
Here ne is the electron density,ni is the ion density, b is the recombination coefficient.
If we set ne = ni = n then integration yields
dn
dt= −bn2
⇒∫ n
n0
dn
n2= −b
∫ t
0
dt
⇒[− 1
n
]nn0
= −bt
⇒ 1
n=
1
n0
− bt
So we have
n =n0
1 + bn0t(4.44)
here n0 is the initial concentration at t = 0 .
Figure 4.2: Process of recombination and Ionisation.
4.9.2 Electron Attachment
During their drift, electrons may be absorbed in the gas by the formation of negative
ions. Whereas the noble gases and most organic molecules can form only stable
negative ions at collision energies of several electron volts (which is higher than the
energies reached during the drift in gas chambers), there are some molecules that are
capable of attaching electrons at much lower collision energies. Such molecules are
sometimes present in the chamber gas as impurities. Among all the elements, the
47
4 Transport of charged particle in Gases
largest electron affinities, i.e. binding energies of an electron to the atom in question,
are found with the halogenides (3.13.7eV ) and with oxygen (∼ 0.5eV ). Therefore
we have in mind contaminations due to air, water, and halogen-containing chemicals.
The break-up of the molecule owing to the attaching electron is quite common and
is called dissociative attachment. The probability of the molecule staying intact is
generally higher at lower electron energies. The rate R of attachment is given by the
cross-section σ , the electron velocity c, and the density N of the attaching molecule:
R = cσN (4.45)
48
Chapter 5
Measurement of IonizationThe moving charges in a chamber give rise to electrical signals on the electrodes that
can be read out by amplifiers. The electrons created in the avalanche close to the
wire move to the wire surface within a time typically much less than a nanosecond,
resulting in a short signal pulse. The ions created in the avalanche move away from
the wire with a velocity about a factor 1000 smaller, which results in a signal with
a long tail of typically several hundred microseconds duration. The movement of
these charges induces a signal not only on the wire but also on the other electrodes
in the chamber, so for the purpose of coordinate measurements the cathode can be
subdivided into several parts. In this chapter we derive very general theorems that
allow the calculation of signals in wire chambers and present some practical examples.
5.1 Signals Induced on Grounded Electrodes, Ramos
Theorem
Contrary to what might be inferred from the brief description of ionization detectors,
the pulse signal on the electrodes of ionization devices is formed by induction due to
the movement of the ions and electrons as they drift towards the cathode and anode,
rather than by the actual collection of the charges itself. A method of computing
the induced current for a charge motion is explained here with the derivation of the
formula using basic electrodynamic identities.
Consider a charge (q), in the presence of any number of grounded conductors, for
one of of any number of grounded conductors, for one of round the electron with a
49
5 Measurement of Ionization
tiny equipotential sphere. Then if V is the potential of the electrostatic field, in the
region between conductors
52V = 0 (5.1)
where 52 is the Laplacian operator. Call Vq the potential of the tiny sphere and note
that V = 0 on the conductors and
−∮S
∂V
∂nds = q/ε0(Gauss
′law) (5.2)
Where the integral is done over a suface of the sphere and ∂V/∂n indicates differenti-
ation with respect to the outward normal to the surface and the integral is taken over
the surface of the sphere. Now consider the same set of conductors with the electron
removed, conductor A raised to unit potential and the other conductors grounded.
Call the potential of the field in this case V ′, so that 52V ′ = 0 in the space between
conductors, including the point where the electron was situated before. Call the new
potential of this point V ′q . Now Green’s theorem states that∮B
[V ′52 V − V 52 V ′
]dv = −
∮B
[V ′∂V
∂n− V ∂V
′
∂n
]ds (5.3)
Here B represents the boundary and ds means that is a surface integral over the
boudary and dv means a volume integral. S stands for the sphere and A is the region
of choosen conductor on which we are calculating the induced charge.
Choose the volume to be that bounded by the conductors and the tiny sphere.
Then the left-hand side is zero and the right-hand side may be divided into three
integrals:
(1) Over the surfaces of all conductors except A. This integral is zero since V = V ′ = 0
on these surfaces.
(2) Over the surface of A. This reduces to −∮A∂V∂nds for V ′ = 1 and V = 0 for
conductor A.
50
5 Measurement of Ionization
(3)Over the surface of the sphere. This becomes −V ′q∮S∂V∂nds+ Vq
∮S∂V ′
∂nds
The second of these integrals is zero by Gauss’ law since∫
(∂V ′/∂n)ds is the negative
of the charge enclosed (which was zero for the second case in which the electron was
removed). Finally, we obtain from (2)
0 = −∮A
∂V
∂nds+ V ′q
∮S
∂V
∂nds = QA/ε0 − qV ′q/ε0
⇒ QA = qV ′q
(5.4)
Now taking the dertivative w.r.t. time we get the induced current;
iA =dQA
dt= q
V ′qdt
= q
[∂V ′q∂x
dx
dt
](5.5)
where x is the direction of the motion.
Now dxdt
= v and∂V ′q∂x
= −Ev, where Ev is electric field parallel to velocity of charge.
so we have,
i = −qvEv (5.6)
5.2 Induced Signals in a Drift Tube
For the cylindrical proportional counter, Fig-5.2 the electric field and potential can
be written as
E(r) =CV02πε
1
r(5.7)
φ(r) = −CV02πε
ln(r/a) (5.8)
where r is the radial distance from the wire, V0 the applied voltage, ε the dielectric
constant of the gas, and
C =2πε
ln(b/a)(5.9)
51
5 Measurement of Ionization
is the capacitance per unit length of this configuration.
Figure 5.1: Cylindrical proportional tube with outer radius b at voltage V0 and inner(wire) of radius a at voltage zero.
Suppose that there is now a charge q located at a distance r from the central wire.
The potential energy of the charge is then the potential energy is W = qφ(r). If now
the charge moves a distance dr, the change in potential energy is
dW = qdφ(r)
drdr (5.10)
For a cylindrical capacitor, however, the electrostatic energy contained in the electric
field is W = l2CV 2, where l is the length of the cylinder. If the movement of the
charges is fast relative to the time that an external power supply can react to changes
in the energy of the system, we can consider the system as closed. Energy is then
conserved, so that
dW = lCV0dV = qdφ(r)
drdr (5.11)
Thus there is a voltage change,
dV =q
lCV0
dφ(r)
drdr (5.12)
induced across the electrodes by the displacement of the charge. Eq.-5.12 is a
general result, in fact, and can be used for any configuration.
52
5 Measurement of Ionization
or our cylindrical proportional counter, let us assume that an ionizing event has
occurred and that multiplication takes place at a distance r′ from the anode. The
total induced voltage from the electrons is then
V − =−qlcV0
∫ a
a+r′
dφ
drdr = − q
2πεlln
(a+ r′
a
)(5.13)
while that from the positive ions is
V + =q
lcV0
∫ b
a+r′
dφ
drdr = − q
2πεlln
(b
a+ r′
)(5.14)
The sum of the two contributions is then V = V − + V + = −q/lC and the ratio of
their contributions is
V −
V +=lna+r
′
a
ln ba+r′
(5.15)
Since the multiplication region is limited to a distance of a few wire radii, it is easy
to see that the contribution of the electrons is small compared to the positive ions.
Taking some typical values of a = 10µm, b = 10mm and r′ = 1µm, V − turns out to
be less than 1% of V +. The induced signal, therefore, is almost entirely due to the
motion of the positive charges and one can ignore the motion of the electrons. With
this simplification we can now calculate the time development of the pulse. Thus,
V (t) =
∫ r(t)
r(0)
dV
drdr = − q
2πεllnr(t)
a(5.16)
To find r(t), considering the mobility to be constant;
dr
dt= µE(r) =
µCV02πε
lnr(t)
a(5.17)
solving the differential equation we get;
r(t) =
(a2 +
µCV0πε
t
)1/2
(5.18)
Now using this expression for the r(t) we can calculate the induced voltage and the
induced charge just by multiplying the capacitance and the time derivative of that
53
5 Measurement of Ionization
will give us the induced current. The wire signal is negative and has a hyperbolic
form with a characteristic time constant t0 , which in practical cases will be a few
nanoseconds. The induced current at time t is given by
I(t) =dQ(t)
dt= Cl
dV (t)
dt=
−q2ln(b/a)
d
dt
[ln
(1 +
t
t0
)]=⇒ I(t) =
−q2ln(b/a)
1
t+ t0
(5.19)
where t0 = a2πε/µCV0.
We could have calculated this easily by using Ramo’s theorem. First calculate
r(t) then the derivative wil give the velocity and then multiplying with the elctric
field we will get the induced current per unit charge.
Figure 5.2: Current signal according to Eq-5.19 (full line, left-hand scale) for t0 =1.25ns, b/a = 500, and q = 106 ∗ e. Time integral of this pulse (Induced charge) as apercentage of the total (broken line, right-hand scale)
Since the electric field in the vicinity of the wire of any wire chamber has the form
1/r, the universal shape of the induced current signal I(t) Eq.-5.19 is valid for wire
signals of all wire chamber geometries.
54
Chapter 6
Simulation of Gaseous detectorsFew definitions to start with:
Model: A system of postulates, data and interfaces presented as a mathematical
description of an entity or proceedings or state of affair. (Development of equations,
constraints and logic rules.)
Simulation: Exercising the model and obtaining results. (Implementation of the
model). Simulation has emerged as the Third methodology of exploring the truth. It
would complement the theory and experimental methodology. Simulation will never
replace them. There are many tangible benefits like Saves manpower, material ; useful
even if not possible by other means; saves money with fast, consistent answers ; could
be used for education after establishing and intangible benifits like increased flexibility,
accuracy, range of operation ; new results not available before ; improved results
due to standardisation ; increased understanding ; explicitly stated assumptions and
constraints in simulations.
The major investments are computer,skill/experties, time for implementation.
There are pitfalls too,
• Modelling errors at different levels -Scientific model of reality, Mathematical
model, discrete numerical model, application program model, computational
model
• Input errors: Out of range inputs can give spurious results
• Precision errors: Limits in the precision
55
6 Simulation of Gaseous detectors
Figure 6.1: An overview of the different methods on quest of truth is explainedthrough a flow chart
6.1 Simulation tools
We are trying to simulate different gaseous detectors using software packages like
GARFIELD, neBEM, MAGBOLTZ.
• GARFIELD : Detector Modelling (specifing the geometry) , Detector Re-
sponse (charge induction using Reciprocity theorem, particle drift, charge shar-
ing, charge collection )
• neBEM : (nearly exact Boundary Element Method)
Electrical Solver: charge distribution on a geometry for a given voltage config-
uration; both potential and field computed using the charge distribution
• MAGBOLTZ : Transport and Amplification
electron drift velocity and diffusion coefficients (longitudinal and transverse),
56
6 Simulation of Gaseous detectors
Townsend and attachment coefficients
6.1.1 Garfield
Garfield, developed by R. Veenhof [12], is a widely used program for detailed sim-
ulation of two and three dimensional drift chambers. The software uses field maps
for these drift chambers as the basis for its calculations. After that, particle drift
including diffusion, avalanches and finally the signal on the read-out electrode can be
calculated.
Garfield provides an efficient two and three dimensional geometry modeler. Dif-
ferent two dimensional (such as infinite equipotential plane, tube) and three dimen-
sional (such as box with right angles, box with a cylindrical hole in center, cylinder,
thin-wire, sphere etc.) elements are defined inside the software. A specific detector
geometry can be modelled with the help of these elements. The description of the
specified chamber can either be in polar or in Cartesian coordinates and consists of
a listing of the position, dimension and potential or dielectric constant.
Garfield has its own library to analytically calculate the electric field when the
detector geometry can be decomposed in equipotential planes, wires and tubes with-
out intersections. For more complicated geometries, the program provides interfaces
with different field solvers.
Considering only the electrostatic fields, the trajectory of an electron or ion would
be such that the position as a function of the time, ~r(t), should obey the following
differential equation
med2~r(t)
dt2= e ~E(~r(t)) (6.1)
where me and e are the mass and the electric charge of the electron. In order to
57
6 Simulation of Gaseous detectors
correctly simulate the drift of the electrons in real gases, Garfield has other integration
methods which employs Monte Carlo calculation that takes diffusion into account.
Finally, Garfield simulates the signal induced on the read-out electrodes resulting
from the passage of a charged particle through the chamber. The electron pulse is
computed by following the avalanche process along the electron drift line. The current
induced by the avalanche ions is also computed according to a simplified model.
6.1.2 MAGBOLTZ
Magboltz was developed by S. Biagi [13] to calculate the transport parameters of
electrons drifting in the gases under the influence of electric and magnetic field.
The program computes electron transport parameters by numerically integrating the
Boltzmann transport equation. By tracking the electron propagation, the program
can compute the drift velocity, the longitudinal and transverse diffusion coefficients
and Townsend and attachment coefficients in various gases. By including a magnetic
field, it can also calculate the Lorentz angle. The collision types involve elastic and
inelastic collisions, attachment, ionization and super-elastic collisions. The collision
angular distributions have also been introduced. The program contains, for 60 gases,
electron cross-sections for all relevant interactions with atoms and/ or molecules. The
gas descriptions are still being improved. In order to improve the simulation and also
maintain the desired accuracy, the Monte Carlo integration technique has been ap-
plied to the solution of the transport equations in the more recent versions of the
code.
•Results using MAGBOLTZ
The theory on transport properties of charges in gases are used with the help of
Monte-Carlo method in MAGBOLTZ. Here some of the gas properties are plotted
against the electric field and the variations are observed which verifies the above
58
6 Simulation of Gaseous detectors
mentioned equations.
Variation of drift velocity in different gas mixtures like Argon-CO2 in different
volume ratios 70-30 , 90-10 and Neon-CO2 in 90-10 and Neon-CO2 − N2 in volume
ratio of 90-10-5 are taken into consideration.
Electric Field (kV /cm)2−10 1−10 1 10 210
sec
)µ
Drif
t Vel
ocity
( c
m/
0
10
20
30
40
50
60
70
80
Ar:CO2-90:10
Ar:CO2-70:30
Ne:CO2-90:10
Ne:CO2:N2-90:10:5
Drift Velocity vs Electric Field
Figure 6.2: Variation of drift velocity of electron in different gas mixtures.
The pairing of lines are completely showing that it is much more dependent on
the base gas like Ar or Ne and the slight difference is due to the change in quenching
gas. Attachment coefficient and Townsend coefficient are plotted which depends on
the in-elastic scattering cross-section of the gases used in the magboltz.
59
6 Simulation of Gaseous detectors
Electric Field (kV /cm)1 10 210
Tow
nsen
d C
oeffi
cien
t (1
/cm
)
0
1000
2000
3000
4000
5000
6000Ar:CO2-90:10
Ar:CO2-70:30
Ne:CO2-90:10
Ne:CO2:N2-90:10:5
Townsend coefficient vs Electric Field
Figure 6.3: Variation of Townsend coefficient of electron in different gas mixtures.
Electric Field (kV /cm)1 10 210
Atta
chm
ent C
oeffi
cien
t (
1/cm
)
0
1
2
3
4
5
6Ar:CO2-90:10
Ar:CO2-70:30
Ne:CO2-90:10
Ne:CO2:N2-90:10:5
Attachment Coefficient vs Electric Field
Figure 6.4: Variation of attachment coefficient of electron in different gas mixtures.
The variation of longitudinal and transverse diffusion coefficient are given below.
60
6 Simulation of Gaseous detectors
The pairing of lines at high electric field value is clear and diffusion coefficients are
same for all low electric field values.
Electric Field (kV /cm)2−10 1−10 1 10 210
)cm
Long
itudi
nal D
iffus
ion
Coe
ffici
ent
(
0.01
0.02
0.03
0.04
0.05
0.06
0.07Ar:CO2-90:10
Ar:CO2-70:30
Ne:CO2-90:10
Ne:CO2:N2-90:10:5
Figure 6.5: Variation of longitudinal diffusion coefficient of electron in different gasmixtures.
Electric Field (kV /cm)
2−10 1−10 1 10 210
)cm
Tra
nsve
rse
Diff
usio
n C
oeffi
cien
t (
0.01
0.02
0.03
0.04
0.05
0.06
0.07 Ar:CO2-90:10
Ar:CO2-70:30
Ne:CO2-90:10
Ne:CO2:N2-90:10:5
Figure 6.6: Variation of Transverse diffusion coefficient of electron in different gasmixtures.
61
6 Simulation of Gaseous detectors
For the variation with the magnetic field values we have taken Ne–CO2–N2 (90-
10-5) and except for the diffusion coefficient there was no variation in any other
parameters. The variation in transverse diffusion coefficient is clear from the eq-4.34
& 4.35.
Electric Field (kV /cm)
2−10 1−10 1 10 210
)cm
Long
itudi
nal D
iffus
ion
Coe
ffici
ent
(
0.01
0.02
0.03
0.04
0.05
0.06
0.07
B = 0T
B = 0.2T
B = 0.4T
B = 0.6T
B = 0.8T
B = 1.0T
Figure 6.7: Variation of longitudinal diffusion coefficient of electron in Ne–CO2–N2
(90-10-5) with magnetic field.
62
6 Simulation of Gaseous detectors
Electric Field (kV /cm)
2−10 1−10 1 10 210
)cm
Tra
nsve
rse
Diff
usio
n C
oeffi
cien
t (
0.02
0.03
0.04
0.05
0.06
0.07B = 0T
B = 0.2T
B = 0.4T
B = 0.6T
B = 0.8T
B = 1.0T
Figure 6.9: Variation of transverse diffusion coefficient of electron in Ne–CO2–N2
(90-10-5) with magnetic field (Perpendicular to electric field).
Electric Field (kV /cm)2−10 1−10 1 10 210
)cm
Tra
nsve
rse
Diff
usio
n C
oeffi
cien
t (
0.02
0.03
0.04
0.05
0.06
0.07 B = 0T
B = 0.2T
B = 0.4T
B = 0.6T
B = 0.8T
B = 1.0T
Figure 6.8: Variation of transverse diffusion coefficient of electron in Ne–CO2–N2
(90-10-5) with magnetic field (parallel to electric field).
The spreading in the low electric field values for the transverse diffusion coefficient
63
6 Simulation of Gaseous detectors
is more when the electric field and magnetic field are parallel and less when they are
perpendicular to each other.(As the Lorentz force is perpendicular to the Magnetic
field applied).
6.1.3 Calculation of the electric field
The process of detailed detector simulation begins with computing estimates of the
electromagnetic field configuration of a given device under experimental conditions.
The efficiency and precision of the field solver determines a large number of parameters
and processes that finally determine the detector characteristics and performance. For
example, the field configuration is a key factor in determining the amount of ionization
and multiplication in a gas. In addition, non-linear effects of space charge, dynamic
charging processes within ionization detectors makes fast but reliable estimation of
electromagnetic fields, a must. While the importance of correctly estimating the field
configuration has always been of major importance in understanding a gas ionization
detector, it is found to be even more difficult and demanding for micro-pattern gas
detectors. This is so because of the large variation in length scales, presence of very
closely spaced surfaces leading to possible degeneracy of boundary conditions, the
importance of resolving various properties and processes to smaller scales, possibility
of unwanted and damaging sparks or discharges and intricate small patterns leading
to large variation in potential and flux within very small distances. The problem is
further complicated through the existence multiple dielectrics in a given device.
The BEM [5], Boundary Element Method , solves the problem only at the bound-
ary elements. If an arbitrary point of interest does not match with a nodal point of
a finite- difference or finite-element mesh, property values at the surrounding mesh
needs to be inter / extrapolated resulting in an obvious loss of accuracy. Since this
method solves for potential and uses low-order polynomials to represent it, the electric
64
6 Simulation of Gaseous detectors
field values are found to be even 50% in error in regions where the fields values are
changing rapidly, e.g., near an anode wire in a proportional counter. It may be noted
here that since a charged particle in a gas detector can occur at any point within the
detector and cause an avalanche, accurate knowledge of electric field at any arbitrary
point within a detector is crucially important to be able to track the behavior of the
charged particle and the avalanche.
Poissons equation for electrostatic potential
∇2φ = −ρ/ε0 (6.2)
can be solved to obtain the distribution of charges which leads to a given potential
configuration φ. Here ρ is the charge density and ε0 is the permittivity of free space.
For a point charge q at ~r′ in 3D space, the potential φ(~r) at ~r is known to be
φ(~r) =q
4πε0|~r − ~r′|(6.3)
For a general charge distribution with charge density ρ(~r′) , superposition holds and
results in
φ(~r) =
∫ρ(~r′)dv′
4πε0|~r − ~r′|=
∫G(~r, ~r′)ρ(~r′)dv′ (6.4)
where
G(~r, ~r′) =1
4πε0|~r − ~r′|(6.5)
is the free space Greens function for the Laplace operator in 3D. Similarly, the field
for a general charge distribution can be written as
~E(~r) = −∇φ (6.6)
so we have
~E(~r) = −∇(∫
G(~r, ~r′)ρ(~r′)dv′)
(6.7)
65
6 Simulation of Gaseous detectors
and, finally
~E(~r) =
∫ρ(~r′)(~r − ~r′)dv′
4πε0|~r − ~r′|3(6.8)
The charge distribution can be obtained from equation (4.3) or (4.7) by satisfying
the boundary conditions at collocation points known either in the form of potential
(Dirichlet) or field (Neumann) or a mixture of these two (Mixed/Robin) on material
boundaries/surfaces present in the domain. Considering the Dirichlet problem only
at present (for ease of discussion), the following integral equation of the first kind can
be set up.
φ(~r) =
∫vol
G(~r, ~r′)ρ(~r′)dv′ (6.9)
Here, φ(~r) is the potential at a point ~r in space and ρ(~r′) is the charge density at
an infinitesimally small volume dv′ placed at , ~r′. The problem is, generally, to find
ρ(~r′) as a function of space resulting the known distribution of φ(~r) . Once the charge
distribution on the boundaries and all the surfaces are known, potential and field at
any point in the computational domain can be obtained using the same equation (4.8)
and its derivative.
The primary step of the BEM technique is to discretize the boundaries and surfaces
of a given problem. The elements resulting out of the discretization process are
normally rectangular or triangular though elements of other shapes are also used.
In the collocation approach, the next step is to find out charge distribution on the
elements that satisfies equation (4.8) following the given boundary conditions. The
charge distribution is normally represented in terms of known basis functions with
unknown coefficients. For example, in zeroth order formulations using constant basis
function, which is also the most popular one among all the BEM formulations because
of a good optimization between accuracy and computational complexity, the charge
distribution on each element is assumed to be uniform and equivalent to a point
66
6 Simulation of Gaseous detectors
charge located at the centroid of the element.
Since the potentials on the surface elements are known from the given potential
configuration, equation (3) can be used to generate algebraic expressions relating
unknown charge densities and potentials at the centroid of the elements. One unique
equation can be obtained for each centroid considering influences of all other elements
including self influence and, thus, the same number of equations can be generated as
there are unknowns. In matrix form, the resulting system of simultaneous linear
algebraic set of equations can be written as follows
K · ρ = φ (6.10)
where K is the matrix consisting of influences among the elements due to unit
charge density on each of them, ρ represents a column vector of unknown charge
densities at centroids of the elements and φ represents known values of potentials at
the centroids of these elements. Each element of this influence coefficient or capacity
coefficient matrix, K is a direct evaluation of an equation similar to equations (4.3) or
(4.7) which represents the effect of a single element on a boundary/surface (obtained
through discretization) on a point where a boundary condition of the given problem
is known. While, in general, this should necessitate an integration of the Greens
function over the area of the element, this integration is avoided in most of the BEM
solvers through the assumption of nodal concentration of singularities with known
basis function.
Since the right hand side of (4.9) is known, in principle, it is possible to solve
the system of algebraic equations and obtain surface charge density on each of the
element used to describe the conducting surfaces of the detector following
ρ = K−1 · φ (6.11)
67
6 Simulation of Gaseous detectors
Once the charge density distribution is obtained, equations (4.3) and (4.7) can be
used to obtain both potential and field at any point in the computational domain.
The nearly exact prefix refers to a way of addressing the singularities near the
boundaries. In the neBEM approach, the integrations for evaluating influences (both
potential and flux) due to rectangular and triangular elements having uniform charge
density have been obtained as closed-form analytic expressions using symbolic math-
ematics. The Inverse Square Law Exact Solutions (ISLES) library is a small library
of C functions based on the analytic closed-form expressions mentioned before. Since
any real geometry can be represented through elements of the above two types (or by
the triangular type alone), this library has allowed us to develop the neBEM solver
that is capable of solving three-dimensional potential problems involving arbitrary
geometry. Its application is not limited by the corners, edges, proximity of other sin-
gular surfaces or their curvature or their size and aspect ratio. It may be noted here
that any non-right-angled triangle can be easily decomposed into two right- angled
triangles. Thus, the right-angled triangles considered here, in fact, can take care of
any three-dimensional geometry.
6.1.4 Accuracy check of neBEM
Calculations for electric field value using neBEM and the theoretically calculated value
for a particular voltage configuration has been carried out. The electric field value
(only magnitude) for a cylindrical proportional chamber comes with an expression
E =V
rln( ba)
(6.12)
For this simple case it can easily be said that electric field is radially outward from
the geometry of the model.
Where b→ inner radius of the cylinder
68
6 Simulation of Gaseous detectors
a→ radius of the central wire
r → radial distance from the axis
V → Potential difference between the electrodes
We have taken a particular configuration with b = 0.8cm , a = 0.005cm and V =
2000 volt Defining the geometry needs a parameter (say n), which defines the number
of sides of the surface of the hole. For example n=2 means the cross section of the
hole will be a polygon with sides 4 (4n-4). So for a perfect circular hole n should be
infinity but then it is time consuming to take a high value of n and do the calculation.
In order to observe the effect from the value of n a comparison has been taken by
two different values of the n and comparing with theoretically calculated values. The
table found out to be
r in cm ETheory ECal (n=4) % error ECal (n=10) % error0.05 7881.50 7921.84 0.512 7888.19 0.0840.1 3940.75 3960.8 0.509 3943.95 0.0810.2 1970.38 1980.16 0.496 1971.7 0.0670.3 1313.58 1319.86 0.478 1314.18 0.0450.7 562.97 586.82 4.236 562.75 0.039
Table 6.1: Comparison of theoretical calculation and neBEM field values.
We can see clearly that the error decreases very rapidly by changing value of n
from 4 to 10. So a moderate value of n was taken for the rest of the calculations.
This shows the accuracy of neBEM with theory.
As an output from Garfield we can get for example the drift lines of the electrons
and ions, a visualisation of the electric field which is calculated from the potential, or
the number of electrons and ions produced in avalanches. This last value is important
69
6 Simulation of Gaseous detectors
to get values for the gain and the ion-backdrift. The same approach is valid for double
and triple GEM stacks with the only difference being that the corresponding basic
cell consists of several single GEM basic cells arranged in a stack. The aim of the
Garfield-simulation is to obtain values for the gain and the Ion-backdrift and to
determine their dependence on different GEM parameters like the GEM voltages or
the distances between the GEMs in the stacks.
70
Chapter 7
Results and DiscussionThe general theory of gaseous ionisation detectors have been used in two different
cases 1)MWPC and 2) GEM
7.1 Multi Wire Proportional Chamber
In 1968, Georges Charpak, while at the European Organization for Nuclear Research
in CERN, invented and developed the multi-wire proportional chamber (MWPC).
The chamber was an advancement of the earlier bubble chamber, rate of detection
of only one or two particles every second to 1000 particle detections every second.
A multiwire proportional chamber (MWPC) consists of a plane of equally spaced
anode wires, sandwiched between two cathode planes. The equipotential lines for
such a structure show three regions of electric field . In the major part of the volume
the field is uniform ; close to the wire it varies inversely with the distance r to the
axis, almost exactly as in a cylindrical counter; and just between the wires there is
a small region of very low field . If a particle produces ion pairs in the gas of such a
structure, the liberated electrons drift towards the anode wire. Reaching its vicinity a
multiplicative avalanche develops. We find the same variety of amplification processes
: an avalanche size proportional to the number of initial electrons for low gains,
an amplification saturated by space-charge effects for high gains, a propagation by
secondary photons for gases with low quenching properties leading either to a Geiger
type propagation along the wires or or to a streamer mode, interrupted or bridging
the electrodes and followed by a spark.
71
7 Results and Discussion
Figure 7.1: Schematic diagram of a MWPC read out and use of gating grid.
In order to get the controls over the charges flowing towards the anode wire a
gating grids are placed in between the anode wire and cathode plane. Using the
GARFIELD the geometry has been created for a single cell which contains a single
wire and later using periodicity it can be expanded to arbitrary number of anode
wires later. The voltage configuration and the electric field intensity along the z-axis
is plotted.
Figure 7.2: Volatage and Electric field configuration of MWPC.
72
7 Results and Discussion
Figure 7.3: Simulating an Avalanche in MWPC.Electron (red line) is drifting from (0,0,0.9) to (0,0,0.2) near which avalanche ishappening and ions(yellow lines) are moving towards the cathode plate Some of
them are getting attached in the gating grid.
7.2 Gas Electron Multiplier
The Gas Electron Multiplier (GEM) detector is another common choice among MPGDs
for tracking and triggering in particle physics experiments. In harsh radiation environ-
ments of high-luminosity colliders, these detectors offer excellent spatial and temporal
resolution. Besides that, the large sensitive area and the operational stability make
this detector a promising candidate in different fields. In the present work, numerical
simulation has been used as a tool of exploration to evaluate the fundamental features
of a single GEM detector.
73
7 Results and Discussion
The Gas Electron Multiplier (GEM) was introduced in 1996 by Sauli [11]. It is a
composite grid structure consisting of two metal layers separated by a thin insulator
which is etched with a regular matrix of holes (Figure 7.4). Applying a voltage
between the two conductive plates, a strong electric field is generated inside the holes
(EGEM in Figure 7.7). It separates the gas volume in three regions: a low field drift
region above the GEM where the primary charge is produced, a high field region
inside the holes where the electrons are multiplied and an induction region below the
GEM where about 50% of the avalanche electrons drift to the read-out electrodes.
7.2.1 Principle of operation
The basic element of the GEM detector is a thin, self-supporting layered mesh realized
by the conventional photo-lithographic methods used to produce multilayer printed
circuits. A thin insulating polymer foil metallized on each side is passivated with
photo-resist and exposed to light through a mask; after curing, the metal is patterned
on both sides by wet etching and serves as self-alignment mask for the etching of the
insulator in the open channels. Because of the etching process, holes are conical
in shape from both entry sides, probably improving the dielectric rigidity. Thanks
to the focusing effect of the field, one expects full efficiency for transfer of charge,
and the dense channel spacing reduces image distortions. For the device to properly
function, a good and regular insulation between the grid electrodes is required, with
no sharp edges, metallic fragments or conducting deposits in the channel; this has
been obtained by careful optimization of the etching and cleaning procedures.
For convenience, the MWPC is operated with the anode wires at positive po-
tentials, the signals being picking up through HV decoupling capacitors; this choice
allows to maintain the lower electrode of GEM at ground potential, and easily increase
the multiplying voltage.
74
7 Results and Discussion
Figure 7.4: Image of the GEM foil taken with an electron microscope.
The standard CERN GEMs consist of an insulator made of a thin Kapton foil
(about 50 µm) which is coated on both sides with copper layers (about 5 µm). This
structure is perforated with holes that typically have a diameter of 70µm and a pitch of
140 µm. The holes are arranged in a hexagonal pattern. Due to an etching production
process they have a double conical shape with an inner diameter of about 50 µm.
Besides the CERN GEMs, also other companies are now producing GEM foils, that
vary in hole size, shape as well as the insulator thickness and material. Figure(7.4)
shows a picture of a GEM that has been taken with an electron microscope. Here
GEM means single GEM at all the considerations.
Most electric field lines end on the side towards the cathode while on the other
side most lines go into the direction of the anode. The ions from the gas amplification
are pulled to and collected on the GEM surface while most of the electrons are
extracted out of the GEM holes. The electron extraction is intensified if additionally
a magnetic field is applied perpendicular to the GEM plane. The electrons tend to
follow the magnetic field lines. The intrinsic ion back-drift suppression is one of the
main advantages of the GEMs and makes a gating grid unnecessary. Operation of
wire chambers typically involved only one voltage setting: the voltage on the wire
75
7 Results and Discussion
provided both the drift field and the amplification field. A GEM-based detector
requires several independent voltage settings: a drift voltage to guide electrons from
the ionization point to the GEM, an amplification voltage, and an extraction/transfer
voltage to guide electrons from the GEM exit to the readout plane. A detector with
a large drift region can be operated as a time projection chamber; a detector with a
smaller drift region operates as a simple proportional counter.
Figure 7.5: Schematic diagram of a GEM detector’s readout.
A GEM chamber can be read-out by simple conductive strips laid across a flat
plane; the readout plane, like the GEM itself, can be fabricated with ordinary lithog-
raphy techniques on ordinary circuit board materials. Since the readout strips are
not involved in the amplification process, they can be made in any shape; 2-D strips
and grids, hexagonal pads, radial/azimuthal segments, and other readout geome-
tries are possible. One notable early user was the COMPASS experiment at CERN.
GEM-based gas detectors have been proposed for components of the International
Linear Collider, the STAR experiment and PHENIX experiment at the Relativistic
Heavy Ion Collider, and others. The advantages of GEMs, compared to multiwire
proportional chambers, include: ease of manufacturing, since large-area GEMs can in
principle be mass-produced, while wire chambers require labor-intensive and error-
76
7 Results and Discussion
prone assembly; flexible geometry, both for the GEM and the readout pads; and
suppression of positive ions, which was a source of field distortions in time-projection
chambers operated at high rates. A number of manufacturing difficulties plagued
early GEMs, including non-uniformity and short circuits, but these have to a large
extent been resolved.
For the GEM-TPC upgrade of ALICE, it is mandatory to minimize the ion back-
flow as a prerequisite for continuous readout and maintenance of excellent TPC per-
formance. GEM foils as a charge amplifier are the candidate to operate a TPC in
continuous readout mode. The GEM technology has been established in the last
decade as a robust and well proved amplification technique for gaseous detectors with
excellent results for spatial resolution, transverse and longitudinal hit separation and
low ion backflow.
The model of a basic GEM cell built using Garfield, is shown in fig- 7.6. It
represents a GEM foil, having two bi-conical shaped holes placed in a staggered
manner along with a readout anode and a drift plane on either sides of the foil. The
distance between top surface of the GEM and the drift plane is called the drift gap
whereas that between the lower surface and the readout plate is named induction
gap. With the help of this model, the field configuration of the detectors has been
simulated using appropriate voltage settings. These are followed by the simulation of
electron transmission and ion backflow fraction in Ne/CO2/N2 (90/10/5) gas mixture.
77
7 Results and Discussion
Figure 7.6: Basic GEM cell built using Garfield.
Figure 7.7: Voltage and Electric field along the z-axis passing through hole of a GEM.
78
7 Results and Discussion
Drift lines:- The electrons, without being affected by the diffusion, follow the
field lines. Due to the high field gradient between the drift volume and the GEM
hole, the field lines are compressed, resulting in a characteristic funnel shape. The
decrease of Egem for a particular Edrift or the increase of Edrift at a fixed Egem , affects
the funneling, resulting in the termination of the drift field line on the top cathode
surface of the GEM foil.
Figure 7.8: Electron drift lines in GEM.
79
7 Results and Discussion
Figure 7.9: A simulated avalanche, drift and the diffusion of secondary electrons(blue) and ions (yellow).
7.2.2 Results
For the calculation of electron transmission, 10000 electrons have been injected in the
drift gap in random position. These electrons are made to drift towards the GEM
foil. The electron transmission has been estimated as the ratio between the number
of electrons that reach the anode plate to the number of electrons created in the drift
volume. For a single GEM detector, the total electron transmission (∈tot) can be
identified as the multiplication of two efficiencies, the collection efficiency (∈coll) and
the extraction efficiency (∈ext).
∈tot = ∈coll × ∈ext (7.1)
80
7 Results and Discussion
The collection efficiency has been defined as:
∈coll =Electrons reached inside the GEM foil
Electrons created in drift volume(7.2)
The extraction efficiency has been defined as:
∈ext =Electrons reached the readout plane
Electrons presented inside the GEM foil(7.3)
The field geometry has a strong impact on ∈coll and ∈ext and thus on ∈tot. A
decrease in the ratio EGEM/EDrift results in the termination of the drift lines on
the top surface of the GEM foil leading to a loss of ∈coll. With a decreasing ratio
EInduction/EGEM , more drift lines are attracted by the bottom surface of the GEM foil
leading to a loss of ∈ext. Again, depending on these two field ratios, some drift lines
also end at the dielectric substrate. In reality, the electron trajectories are affected
by the diffusion and the loss of electrons on different electrodes increases due to
diffusion which naturally affects ∈coll and ∈ext. Besides that, the electron attachment
coefficient, can also influence transmission. The variations of ∈coll, ∈ext and ∈tot under
different field configurations have been plotted in following figures.
For a fixed VGEM and EInduction, ∈coll and thus ∈tot, decrease with the increase of
the drift field, whereas no significant effects of drift field on ∈ext has been observed
(figure). Similarly, at a fixed VGEM and EDrift, the increase of induction field, in-
creases ∈ext as shown in fig 7.11. The change of VGEM only has effect on ∈coll and
thus ∈tot (figure). It is also seen from fig 7.12, for the same voltage configuration, the
smaller pitch GEM foils is better in terms of higher electron transmission, whereas
no significant effect of 0.5 T magnetic field has been observed.
81
7 Results and Discussion
[V/cm]DriftE310
tot
∈,ex
t∈,
coll
∈
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8coll∈
ext∈
tot∈
90/10/52/N2
= 4000 V/cm , Ne/COinduction
= 350 V , EGEM V
Figure 7.10: Variation of electron transmission and efficiencies as a function of driftfield.
[V/cm]InductionE310
tot
∈,ex
t∈,
coll
∈
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
coll∈
ext∈
tot∈
90/10/52/N2
= 400 V/cm , Ne/CODrift
= 350 V , EGEM V
Figure 7.11: Variation of electron transmission and efficiencies a function of inductionfield.
82
7 Results and Discussion
[V]GEMV200 250 300 350 400 450
tot
ε
0.15
0.2
0.25
0.3
0.35
0.4
0.45m (S), B = 0TµPitch: 140
m (S), B = 0.5TµPitch: 140m (LP), B = 0TµPitch: 280
90/10/52/N2
= 4000 V/cm , Ne/COinduction
= 400 V/cm , EDrift E
Figure 7.12: Variation of electron transmission and efficiencies a function of GEMvoltage (For different pitch).
The electrons during their drift produce avalanche inside the GEM foil. The
primary ions in the drift region and the ions created in the avalanche have been
considered for the estimation. The backflow fraction has been calculated as
IBF =Nid
NiT
(7.4)
where Nid is the number of ions collected at the drift plane and NiT is the number
of total ions. As mentioned earlier, the ions drifting back to the drift volume, can
disturb the homogeneity of the drift field and, thus, distort the behaviour of the
detector. In order to prevent those ions from entering the drift volume, a proper
optimization of the field in the drift volume, GEM hole and induction regions is
necessary (figure 7.13).
83
7 Results and Discussion
finalEntries 0Mean 0RMS 0
[V/cm]Induction
,EDriftE310 410
Ion
Bac
kflo
w F
ract
ion
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
finalEntries 0Mean 0RMS 0
= 4000 V/cmInduction
, EDriftE
= 400 V/cmDrift
, EInductionE
(90/10/5)2
/N2
m , Ne/COµ = 350 V , Pitch = 140 GEMV
Figure 7.13: Variation of IBF a function of drift and induction field.
The ion backflow of a single GEM can be reduced by decreasing EDrift because
less number of field lines will get out of the hole into the drift volume. At higher
EDrift, the ratio between and EGEM is large resulting in the drift of more number of
ions into the drift volumm. At higher EGEM, the ratio between EDrift and EGEM is
small and thus a large fraction of ions is collected at the top surface of the GEM foil.
No significant effect of EInduction has been observed except at the higher EInduction.
From the figure 7.14, it is also seen that the GEM foil with smaller pitch is better in
terms of lower backflow fraction, whereas no significant effect of 0.5T magnetic field
on backflow has been observed.
84
7 Results and Discussion
[V]GEMV200 250 300 350 400 450
Ion
Bac
kflo
w F
ract
ion
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6 mµPitch: 140mµPitch: 280
90/10/52/N2
= 4000 V/cm , Ne/COinduction
= 400 V/cm , EDrift E
Figure 7.14: Variation of IBF as a function of GEM voltage (For different pitch).
The Avalanche routine of Garfield software has been used. The procedure first
drifts an initial electron from the specified starting point. At each step, a number
of secondary electrons is produced according to the local Townsend and attachment
coefficients and the newly produced electrons are traced like the initial electrons.
Changing the three voltage differences the variation of GEM gain is checked. Gain
is plotted below with varying GEM voltage in a log scale. Then it has been fitted
with a exponential function.
85
7 Results and Discussion
GEM Voltage (volt)250 300 350 400 450
Gai
n
10
210
/ ndf 2χ 3.777 / 3
Constant 0.1526±4.5 −
Slope 0.0003446± 0.02131
/ ndf 2χ 3.777 / 3
Constant 0.1526±4.5 −
Slope 0.0003446± 0.02131
Gain vs Voltage
Figure 7.15: Variation of Gain as a function of GEM Volatge.
Which shows that the avalanche process amplifies more with increasing electric
field between the GEM plates.
7.2.3 Conclusion
As the increment in voltage across the GEM foil increases the gain,electron trans-
mission and decreases the IBF, high GEM voltage should be prefered. Lower pitch
(140µm) and low drift field allows low IBF values and high electron transmission.Induction
field should be high as it increases the electron transmission.
86
Chapter 8
Summary and ConclusionsWe have discussed the principles of the operation of the gaseous detectors focusing
on the most relevant mechanisms involved in the detection process. This knowledge
is necessary to understand and optimize the performance of a proposed or any exist-
ing detection systems. In our work we have used the Garfield simulation framework
for carrying out the simulation of the physical processes in different MPGDs. This
framework was augmented in 2009 through the addition of the neBEM toolkit to
carry out 3D electrostatic field simulation. Besides neBEM, the Garfield framework
provides interfaces to Magboltz for computing the transport and amplification prop-
erty. We have discussed about the these simulation tools. To understand the software
in details, we have also presented some useful results using these tools as stand-alone
routines. Starting with the discussion of an MWPC, we have emphasized our focus
on the MPGDs. Among several MPGDs, we have given a somewhat detailed descrip-
tion, advantages and disadvantages of GEM detector. In the present work, numerical
simulation has been used as a tool of exploration to evaluate the fundamental features
of a single GEM detector. The study includes extensive computation of electrostatic
field configuration within a given device. Some of the fundamental properties like
gain, electron transmission, ion backflow and their dependence on different voltage
configurations, have been estimated too.
87
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89